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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I HARVARD UNIVERSITY PHYSICS RESEARCH LIBRARY Gift of J. H. Van Vleck (^j^.^KwTl A COURSE OF MODERN ANALYSIS lonlion: C. J. OLAY and SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AYE MARU LANE. •UuwoiD: 60, WELLINGTON STREET. lrtp}tg: F. A. BBOCKHAUS. ^ciD «odt: THE MAOMILLAN GOMPANT. SombSB and ColnitU: MAOMILLAN AND 00.. Ltd. [All RighU reserved.] lS7T>A(.Z, A COURSE OF MODERN ANALYSIS AN INTRODUCTION TO THE GENERAL THEORY OF INFINITE SERIES AND OF ANALYTIC FUNCTIONS; WITH AN ACCOUNT OF THE PRINCIPAL TRANSCENDENTAL FUNCTIONS BY K T. WHITTAKER, M.A, FELLOW AND LECTURER OF TRINITY COLLEGE, CAMBRIDGE CAMBRIDGE: AT THE UNIVERSITY PRESS. 1902 loo Cambrttgr : PRINTBD BT J. AHD 0. F. CLAT, AT THE UNIVBR8ITT PRR88. Physics Research Librar> Jefferson Laboratoiy Harvard University MAR 19 1982 -?"'- ' V y PREFACE. The first half -of this book contains an account of those methods and processes of higher mathematical analysis, which seem to be of greatest importance at the present time ; as will be seen by a glance at the table of contents, it is chiefly concerned with the properties of infinite series and complex integrals, and their applications to the analytical expression of functions. A discussion of infinite determinants and of asymptotic expansions has been included, as it seemed to be called for by the value of these theories in connexion with linear differential equations and astronomy. In the second half of the book, the methods of the earlier part are applied in order to furnish the theory of the principal functions of analysis — the Gamma, Legendre, Bessel, Hypergeometric, and Elliptic Functions. An account has also been given of those solutions of the partial differential equations of mathematical physics, which can be constructed by the help of these functions. My grateful thanks are due to two members of Trinity College, Rev. E. M. Radford, M.A. (now of St John's School, Leatherhead), and Mr J. E. Wright, B.A., who with great kindness and care have read the proof-sheets; and to Professor Forsyth, for many helpful consultations during the progress of the work. My great indebtedness to Dr Hobson*s memoirs on Legendre functions must be specially mentioned here; and I must thank the staff of. the University Press for their excellent co- operation in the production of the volume. E. T. WHITTAKER. Cambridge, 1902 August 5 CONTENTS. PART I. THE PROCESSES OF ANALYSIS. CHAPTER I. COMPLEX NUMBERS. 8KCTI0N PAGE 3 4 1. Heal numbers 2. Complex uumbers 3. The modulus of a complex quantity 5 4. The geometrical interpretation of complex numbers 6 Miscellaneous Examples 7 CHAPTER II. THE THEORY OF ABSOLUTE CONVERGENCE. 5. The limit of a sequence of quantities 8 6. The necessary and sufficient conditions fi)r the existence of a limit . . 8 7. Convergence of an infinite series 10 8. Absolute convergence and semi-convergence 12 9. The geometric series, and the series 2n~* 13 10. The comparison-theorem 14 11. Discussion of a special series of importance 16 12. A convergency-test which depends on the ratio of the successive terms of a series 17 13. A general theorem on those series for which Limit ( -* ) is 1 . . . 18 n-ao \u^ J 14. Convergence of the hypergeometric series 20 15. Effect of changing the order of the terms in a series 21 16. The fundamental property of absolutely convergent series .... 22 17. Riemann's theorem on semi-convergent series 22 18. Cauch/s theorem on the multiplication of absolutely convergent series . . 24 19. Mertens' theorem on the multiplication of a semi-convergent series by an absolutely convergent series 25 20. Abel's result on the multiplication of series 26 21. Power-series 28 X CONTENTS. SECTION PAGE 22. Convergence of series derived from a power-series 30 23. Infinite products 31 24. Some examples of infinite products 32 25. Cauchy's theorem on products which are not absolutely convergent . . 34 26. Infinite determinants 35 27. Convergence of an infinite determinant 36 28. Persistence of convergence when the elements are changed .... 37 Miscellaneous Examples 37 CHAPTER III. THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS; TAYLOR'S, LAURENT'S, AND LIOUVILLE'S THEOREMS. 29. The dependence of one complex number on another 40 30. Continuity 41 31. Definite integrals 42 32. Limit to the value of a definite integral 44 33. Property of the elementary functions 44 34. Occasional failure of the property ; singularities 45 35. The analytic function 45 36. Cauchy's theorem on the integral of a Amotion round a contour . . 47 37. The value of a function at a point, expressed as an integral taken round a contour enclosing the point 50 38. The higher derivates 51 39. Taylor's theorem 54 40. Forms of the remainder in Taylor's series 56 41. The process of continuation 57 42. The identity of a function 59 43. Laurent's theorem 60 44. The nature of the singularities of a one-valued function .... 63 45. The point at infinity 64 46. Many-valued functions 66 47. Liouville's theorem 69 48. Functions with no essential singularities 69 'Miscellaneous Examples 70 CHAPTER IV. THE UNIFORM CONVERGENCE OF INFINITE SERIES. ■ 49. Uniform convergence 50. Connexion of discontinuity with non-uniform convergence 51. Distinction between absolute and uniform convergence 52. Condition for uniform convergence 53. Integration of infinite series . 54. Differentiation of infinite series 55. Uniform convergence of power-series Miscellaneous Examples 73 76 77 78 78 81 81 82 CONTENTS. XI CHAPTER V. THE THEORY OF RESIDUES; APPLICATION TO THE EVALUATION OF REAL DEFINITE INTEGRALS. SECTION PAGE 56. Residues 83 57. Evaliiation of real definite integrals 84 58. Evaluation of the definite integral of a rational function .... 91 59. Cauchy's int^ral 92 60. The number of roots of an equation contained within a contour . . 92 61. Connexion between the zeros of a function and the zeros of its derivate . 93 MiSCSLLAKBOUS EXAMPLES 94 CHAPTER VI. THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 62. Darboux's formula 96 63. The Bemoullian numbers and the Bemoullian polynomials ... 97 64. The Maclaurin-Bemoidlian expansion ........ 99 65. Burmann's theorem 100 66. Teixeira's extended form of Burmann's theorem 102 67. Evaluation of the coefficients 103 68. Expansion of a function of a root of an equation, in terms of a parameter occurring in the equation 105 69. Lagrange's theorem 106 70. Rouch^'s extension of Lagrange's theorem 108 71. Teixeira*s generalisation of Lagrange's theorem 109 72. Laplace's extension of Lagrange's theorem 109 73. A further generalisation of Taylor's theorem 110 74. The expansion of a function as a series of rational functions . . . Ill 75. Expansion of a function as an infinite product 114 76. Elxpansion of a periodic function as a series of cotangents . . . . 116 77. Expansion in inverse factorials 117 Miscellaneous Examples 119 CHAPTER VII. FOURIER SERIES. 78. Definition of Fourier series ; nature of the region within which a Fourier series converges 127 79. Values of the coefficients in terms of the sum of a Fourier series, when the series converges at all points in a belt of finite breadth in the «-plane . . 130 80. Fourier's theorem 131 81. The representation of a function by Fourier series for ranges other than to 27r 137 82. The sine and cosine series 138 83. Alternative proof of Fourier's theorem 140 84. Nature of the convergence of a Fourier series 147 85. Determination of points of discontinuity 161 86. The uniqueness of the Fourier expansion 162 Miscellaneous Examples 167 XU CONTENTS. CHAPTER VIII. ASYMPTOTIC EXPANSIONS. SECTION PAOK 87. Simple example of an asymptotic expansion 163 88. Definition of an asymptotic expansion 164 89. Another example of an asymptotic expansion 165 90. Multiplication of asymptotic expansions 167 91. Integration of asymptotic expansions 168 92. Uniqueness of an asymptotic expansion 168 Miscellaneous Examples 169 PART II. TRANSCENDENTAL FUNCTIONS. CHAPTER IX. THE GAMMA-FUNCTION. 93. Definition of the Qamma-f unction, Euler's form 173 94. The Weierstrassian form for the Gamma-function . . . . . 174 95. The difference-equation satisfied by the Gamma-function . . . 176 96. Evaluation of a general class of infinite products 177 97. Connexion between the Gamma-fimction and the circular functions . 179 98. The multiplication-theorem of Gauss and Legendre 179 99. Expansions for the logarithmic derivates of the Gamma-function . . 180 100. Heine*s expression of F («) as a contour-integral 181 101. Expression of T(z) as a definite integral, whose path of integration is real 183 102. Extension of the definite-integral expression to the case in which the argument of the Gamma-fimction is negative 184 103. Gauss' expression of the logarithmic derivate of the Gamma-fimction as a definite integral 185 104. Binet's expression of log F (e) in terms of a definite integral . . 186 105. The Eulerian int^ral of the first kind 189 106. Expression of the Eulerian integral of the first kind in terms of Gkunma- functions 190 107. Evaluation of trigonometric integrals in terms of the Gamma-function 191 108. Dirichlet's multiple integrals 191 109. The asymptotic expansion of the logarithm of the Gamma-fiinction (Stirling's series) 193 110. Asymptotic expansion of the Gamma-fimction 194 Miscellaneous Examples 195 CONTENXa xm CHAPTER X. LEGENDRE FUNCTIONS. BKCTION PAGE 111. DefinitioD of Legendre polynomiab 204 112. Schlafli's integral for P„ ^ 206 113. Rodrigues' formula for the Legendre polynomials 206 114. Legendre's differential equation 206 115. The int^^ral-properties of the L^endre polynomials 207 116. Legendre functions 208 117. The recurrence-formulae 210 118. Evaluation of the integral-expression for P^{z\ as a power-series . . 213 119. Laplace's iut^pral-expression iot P^{z) 215 120. The Mehler-Dirichlet definite integral for /*„(«). . % . . . 218 121. Expansion of P^ {z) as a series of powers of 1/z 220 122. The Legendre functions of the second kind 221 123. Expansion of Q^isi) as a power-series 222 124. The recurrence-formulae for the Legendre function of the second kind . . 2^4 125. Laplace's int^;ral for the Legendre function of the second kind . . . 225 126. Relation between Pm(^) ^^^ Q%{^\ when n is an integer . . . 226 127. Expansion of {t-x)~^ as a series of Legendre polynomials . . 228 128. Neumann's expansion of an arbitrary function as a series of Legendre polynomials 230 129. The associated functions P^ (z) and $»"•(«) 231 130. The definite integrals of the associated Legendre functions . . . 232 131. Expansion of P%^{z) as a definite integral of Laplace's type . . 233 132. Alternative expression of P^ {z) as a definite integral of Laplace's type . 234 133. The function C/ («) 235 MiSCBLLANBOUS EXAMPLES 236 CHAPTER XI. HYPERGEOMETRIC FUNCTIONS. 134. The hypergeometric series 240 136. Value of the series /'(a, 6, c, 1) . 241 136. The differential equation satisfied by the hypergeometric series 242 137. The differential equation of the general hypergeometric function 242 138. The L^^ndre functions as a particular case of the hypergeometric function . 245 139. Transformations of the general hypergeometric function .... 246 140. The twenty-four particular solutions of the hypergeometric differential equation 249 141. Relations between the particular solutions of the hypergeometric differential equation 251 142. Solution of the general hypergeometric differential equation by a definite int^;ral 253 143. Determination of the integral which represents P(*) 257 144. Evaluation of a double-contour int^;ral 259 145. Relations between contiguous hypergeometric functions .... 260 MiSCBLLANBOUS EXAMPLES 263 ; xiv CONTENTS. CHAPTER Xll. BESSEL FUNCTIONS. SECTION PAGE 146. The Bessel coefficients 266 147. Bessel's differential equation 268 148. Bessel's equation as a case of the hjpergeometric equation . . 269 149. The general solution of Bessel's equation by Bessel functions whose order is not necessarily an integer 272 150. The recurrence-formulae for the Bessel functions 274 151. Relation between two Bessel functions whose orders differ by an integer . 275 152. The roots of Bessel functions 277 153. Expression of the Bessel coefficients as trigonometric integrals . . 277 154. Extension of the integral-formula to the caae in which n is not an integer . 279 155. A second expression of J^^ (z) as a definite integral whose path of integration is real 282 156. Hankel's definite-integral solution of Bessel's differential equation . . 283 157. Expression of J^^ (2), for all values of n and «, by an integral of Hankel's type 284 158. Bessel functions as a limiting case of Legendre functions .... 287 159. Bessel functions whose order is half an odd integer 288 160. Expression of J^ (z) in a form which furnishes an approximate value to J^^ (z) for large real positive values of z 289 161. The asymptotic expansion of the Bessel functions 292 162. The second solution of Bessel's equation when the order is an integer . . 294 163. Neumann's expansion ; determination of the coefficients .... 299 164 Proof of Neumann's expansion 300 165. Schl5milch's expansion of an arbitrary functiou in terms of Bessel functions of order zero 302 166. Tabulation of the Bessel functions 304 Miscellaneous Examples 304 CHAPTER XIIL APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 167. Introduction : illustration of the general method 309 168. Laplace's equation ; the general solution ; certain particular solutions . 311 169. The series-solution of Laplace's equation 314 170. Determination of a solution of Laplace's equation which satisfies given boundary-conditions 315 171. Particular solutions of Laplace's equation which depend on Bessel functions 317 172. Solution of the equation ^ -I- ^jj-hr=0 318 92 ^ 92 7 32 y 173. Solution of the equation g^ + ^j-h-^ -I- r=0 319 Miscellaneous Examples 321 CONTENTS. XV CHAPTER XIV. THE ELLIPTIC FUNCTION ^(z), 8BCTI0N PAGE 174. Introduction 322 175. Definition of jf> («) 323 176. Periodicity, and other properties, of ^(z) 324 177. The period-parallelograms 324 178. Expression of the function fp (z) by means of an integral .... 325 179. The homogeneity of the function i> (z) 329 180. The addition- theorem for the function ip{z) 329 181. Another form of the addition theorem 332 182. The roots Ci, eg, 63 333 183. Addition of a half-period to the aigument o{ jp{z) 334 181 Integration of (cw:* + 46a73+6c^ + 4ci:a7+e)-* 336 185. Another solution of the integration-problem 336 186. Uniformisation of curves of genus unity 338 Miscellaneous Examples 340 CHAPTER XV. THE ELLIPTIC FUNCTIONS sn z, en z, dn z, 187. Construction of a doubly-periodic function with two simple poles in each period-parallelogram 342 188. Expression of the function f{z) by means of an integral .... 343 189. The function sn^ 345 190. The functions cm and dn^ 346 191. Expression of en z and dn z by means of integrals 347 192. The addition-theorem for the function dnz 348 193. The addition-theorems for the functions sn z and en ^ . . . 350 191 The constant K 351 195. The periodicity of the elliptic functions with respect to ^ . . 351 196. The constant K' 352 197. The periodicity of the elliptic functions with respect to K+iK' . . 353 198. The periodicity of the elliptic functions with respect to tX' . . . 353 199. The behaviour of the functions snr, en-?, dnz, at the point z=iK' . . 354 200. (General description of the functions sn z, en 2, dn z 355 201. A geometrical illustration of the functions snz, cnz, dnz. . . 356 202. Connexion of the function snz with the function ^{z) . . . . 356 203. Expansion of snz as a trigonometric series 357 Miscellaneous Examples 359 XVI CONTENTS. CHAPTER XVI. ELLIPTIC FUNCTIONS; GENERAL THEOREMS. SBCnON PAQK 204. Relation betweeu the residues of an elliptic function .... 362 205. The order of an elliptic Amotion 362 206. Expression of any elliptic function in terms of fp {z) and fP' (z) . . 363 207. Relation between any two elliptic fimctions which admit the same peri<xls . 364 208. Relation between the zeros and poles of an elliptic function . . 365 209. The function f (2) 366 210. The quasi-periodicity of the function ((z) 367 211. Expression of an elliptic function, when the principal part of its expansion at each of its singularities is given 367 212. The function a{z) 368 213. The quasi-periodicity of the function ^{z) 369 214. The integration of an elliptic function 372 215. Expression of an elliptic function whose zeros and poles are known . 372 Miscellaneous Examples 374 Index 377 J PART I. THE PROCESSES OF ANALYSIS. W. A. CHAPTER I. Complex Numbers. 1. Real Numbers. The idea of a set of numbers is derived in the first instance ifrom the consideration of the set of positive integral numbers, or positive integers ; that is to say, the numbers 1, 2, 3, 4, .... Positive integers have many properties, which will be found in treatises on the Theory of Integral Numbers; but at a very early stage in the development of mathematics it was found that they are inadequate to express all the quantities occurring in calculations ; and so this primitive number system has come to be enlarged. In elementary Arithmetic, and in the arithmetical applications of Algebra, several new classes of numbers are defined, namely rational fractions such as ^, negative numbers such as —3, and irrational numbers such as the number 1*414213..., which represents the square root of 2. The object of the introduction of these extended types of number is that we may express the result of performing the operations of addition, subtraction, multiplicatioo, division, involution, and evolution, on all integral numbers. Thus, the result of dividing the integer 1 by the integer 2 is inexpressible until we introduce the idea of fractional numbers: and the result of subtracting the integer 2 from the integer 1 is inexpressible until we introduce the idea of negative numbers. The totality of the numbers introduced up to this point is called the aggregate of real numbers. The extension of the idea of number, which has just been described, was not effected without some opposition from the more conservative mathematicians. In the latter half of the 18th centiuy, Maseres (1731—1824) and Frend (1767—1841) published works on Algebra, Trigonometry, etc., in which the use of negative quantities was disallowed, although Descartes had used them imrestrictedly more than a hundred years before. 1—2 4 THE PROCESSES OF ANALYSIS. [CHAP. L 2. Complex Numbei^s*. If we attempt to perform the operations already named — multiplication, etc. — on any of the real numbers thus recognised, we find that there is one case in which the result of the operation cannot be expressed without the introduction of yet another type of numbers. The case referred to is that in which the operation of evolution is applied to a negative number, e.g. to find the square root of — 2. To express the results of this and similar opera- tions, we make use of a new number, deooted by the letter t; this is defined as a quantity which satisfies the fundamental laws of algebra (Le. can be combined with other numbers according to the associative, distributive, and commutative laws) and has for its square the negative number — 1. It is easily seen that all the quantities \\h\c\\ can be formed by com- bining i with real numbers are of the form a + 6t, where a and 6 are real numbers. A quantity a + 6i of this nature is called (after Gauss) a complex nv/mber. Real numbers may be regarded as a particular case of complex numbers, corresponding to a zero value of the quantity 6. The complex quantity thus introduced may in the first instance be regarded as formed by the association of the pair of real numbers a and 6; as the quantities a, 6, i are subject to the ordinary laws of algebra, we obtain for the addition and multiplication of two complex numbers a + hi and c + dt the formulae (a + hi) + (c + di) = (o + c) +(6 + d)t, (a + 6t) (c + di) = (oc — bd) + {ad + he) i. But a complex number will usually be considered apart from its composition^ as an irresoluble entity. Regarded in this light, it satisfies the fundamental laws of algebra ; so that if a, 6, c are complex numbers, we have a + 6 a 6 + a, ah = 6a, {a-{'h)-{'C^a-{'{h-\'C\ ah ,c^a ,hc, a (6 + c) = oi + oc. It is found that the operations of multiplication, etc., when applied to complex numbers, do not lead to numbers of any fi"esh type ; the complex number will therefore for our purposes be taken as the most general type of number. The introduction of the complex number has led to many important developments in mathematics. Functions which, when real variables only * For the general theory of complex numbers, see Hankel, Theorie der eomplexen Zahlen- systeme (Leipzig, 1867), and Stolz, VarUsungen Uber allgemeine Arithmetik IL (Leipzig, 1886). 2, 3] COMPLEX NUMBERS. 5 are considered, appear as essentially distinct, are seen to be connected when complex variables are introduced : thus the circular functions are found to be expressible in terms of exponential functions of a complex argument, by the equations cos 07 = 2 («^ + e-^), sin a? = 2i (^** - ^"**)- Again, many of the most important theorems of modem analysis are not true if the quantities concerned are restricted to be real; thus, the theorem that every algebraic equation of degree n has n roots is true in general only when complex values of the roots are admitted. Hamilton's quaternions furnish an example of a still further extension of the idea of number. A quaternion is formed from four real numbers Wy x, y, z, and four number-units 1, i^ j, k, in the same way as the ordinary complex number .r+ty is formed from two real numbers ^, y, and two number-units 1, i Quaternions however do not obey the commutative law of multiplication. 3. The modulus of a complex quantity. Let x+iy he & complex quantity; x and y being real numbers. Then the positive square root of a^ + y^ 18 called the modulus of (x + yi), and is written \x + yi\. Let us consider the complex number which is the sum of two known complex numbers, a? + iy and u + iv. We have (x + iy) + (u + iv) « (a? + w) + i (y + v). The modulus of the sum of the two numbers is therefore or {(aj« + y«) + (u» + v») + 2 (xu -h yv)]K But = (^ + y*) + (t*» + v") + 2 {(xu -^ yvY •¥ (xv - yu)«}*, and this latter expression is greater than (or at least equal to) («" + y*) + (w' + i^)+2(xu + yv). We have therefore I r + iy I + 1 u + iy I > |(a7 + iy) + (u + iv) |, or the modulus of the sum of two complex numbers cannot be greater than the sum of their moduli; and in general it follows that the modulus of the sum 6 THE PROCESSES OF ANALYSIS. [CHAP. I. of any number of complex quantities cannot be greater than the sum of their moduli. Let us consider next the complex number which is the product of two known complex numbers x -k-iy and u + %v; we have {x + iy) {u + iv) = {xu — yv) + 1 (aw + yu), and therefore I {x-\-iy){u + iv) I = [{xu - yvf + (a?w+ ye^)'}* ^\x-^iy\ \u'\-iv\. The modulus of the prodiLCt of two complex quantities (and hence of any number of complex quantities) is therefore equal to the product of their modvli. 4. The geometrical interpretation of complex numbers. For many purposes it is useful to represent complex numbers by a geometrical diagram, which may be done in the following way. Take rectangular axes Ox, Oy, in a plane. Then a point P whose coordinates referred to these axes are x, y, will be regarded as representing the complex number a?+ iy. In this way, to every point of the plane there corresponds some complex number; and conversely, to every possible complex number there corresponds one and only one point of the plane. The complex number x + iy may be denoted by a single letter z. The point P is then called the representative point or affix of the value z ; we shall also speak of the number z as being the affia of the point P. If we denote (a^+ y*)* by r and tan~* (^ i by d, then r and are clearly the radius vector and vectorial angle of the point P, referred to the origin and axis Ox, The representation of complex quantities thus afforded is often called the Argand diagram*. If Pi and Pa are the representative points corresponding to values Zi and z^ respectively of z, then the point which represents the value Zi-{- z^ is clearly the terminus of a line drawn fix)m Pi, equal and parallel to that which joins the origin to Pj. To find the point which represents the complex number z^z^, where Zx and z^ are two given complex' numbers, we notice that if z^ = ri (cos ^1 + i sin d,), iTj = r, (cos ^a + 1 sin d,), ♦ J. R. Argand published it in 1806 ; it bad bowever previously been used by Gauss, and by Caspar Wessel, who discussed it in a miemoir published in 1797 to the Danish Academy. 4] COMPLEX NUMBERS. 7 then by multiplication z^Zi =s r^ra {cos (d, + 0^) + i sin (0i + ^,)}. The point which represents the value ZiZ^ has therefore a radius vector measured by the product of the radii vectores of Pi and Pa, and a vectorial angle equal to the sum of the vectorial angles of Pi and P,. Miscellaneous Examples. 1. Shew that the representative points of the complex numbers 1+4^ 2+7i, 3 + lOt, areooUinear. 2. Shew that a parabola can be drawn to pass through the representative points of the complex numbers 2+t, 4+4i, 6 + 9t, 8+16t, 10+26i. 3. Determine by aid of the Argand diagram the nth roots of unity ; and shew that the number of primitive roots (roots the powers of each of which give all the roots) is the number of integers including unity less than n and prime to it. Prove that if ^i, ^s* ^s* ••• ^ the argmnents of the primitive roots, 2coBpS=0 when p is A positive integer less than -j- — i , where a, b, c^„.k are the different constituent primes of n; and that, when p=-j- — .> 2co6jp^«» , , , where ii is the number of the constituent primes. (Cambridge Mathematical Tripos, Part I. 1895.) CHAPTER II. The Theory of Absolute Convergence. 6. The limit of a sequence of quantities. Let Zi, z^i Zz, ... be a sequence of quantities (real or complex), ififinite in number. The sequence is said to tend towards a limiting value or limit I, provided that, corresponding to every positive quantity €, however small, a number n can be chosen, such that the inequality I Z^ - Z I < 6 is true for all values of m greater than w. If £: is a variable quantity which takes in succession the values ^i, ^, ^t> ••• » then z is said to tend to the Ivmit L Example. Consider the sequence of numbers ^, ^, |)>-<9 for which ^=oii* This sequence tends to the limiting value 1=0; for if any positive quantity c be taken, and if n denote the integer next greater than - y^~o > ^^^^ ^^^ inequality 1 is true for all values of m greater than n. 6. The necessary and sufficient condition for the existence of a limit We shall now shew that the necessary and sufficient condition for the existence of a limiting value of a sequence of finite numbers Zi, z^, ^,, ... is that corresponding to any given positive quantity e, however small, it shall be possible to find a number n such that the equation is verified for aXl positive integral values of p. This may be expressed in words by the statement that a finite variable quantity has a limit if and only if its oscillations have the limit zero ; it may be regarded as one of the fundamental theorems of analysis. First, we have to shew that this condition is necessaiy, i.e, that it is satisfied whenever a limit exists. Suppose then that a limit I exists ; then 5, 6] THE THEORY OF ABSOLUTE CONVERGENCK 9 (§ 5) corresponding to any positive quantity €, however small, a number n can be chosen such that and I Zn^ — Z I < ^ , for all values of p ; therefore I ^n+p — Zn\<\ (Zn+p — ~ (^n — I <\Zf^-l\ + \Zn-l\ € € which shews the necessity of the condition I '^«-H> ■" -^n I < ^1 and thus establishes the first half of the theorem. Secondly, we have to shew that this condition is sufficient, i.e. that if it is satisfied, then a limit exists. Suppose then that this condition is satis- fied Let Zr = a?r + tyr, where Xr and iyr are the real and imaginary parts of Zr. Then if I Zf^p — 2r„ I < €, we have | {x,^ - x^) + i(yn+i> -yn)\<^ and therefore x^ — €< Xn^ < ^n + €, and yn-€< yn+p < yn + e. Now the number n is determined by the quantity €, which can be assigned arbitrarily. Let ni> ^ ^i ^4> ••• be the numbers which correspond in this € € € € way to the quantities ^, j, -, =^,.... Let w* be the least of the quantities Xn + e, Xn^ + 5, ^n, + 7 , ••• ^n» + o^ , SO that the quantities w©, v^, ti„ ... are a decreasing sequence ; and let v^ be the greatest of the quantities € € € so that the quantities Vo, Vi, «,,... are an increasing sequence ; and clearly Then any of the numbers in the li-sequence is greater than any of the numbers in the v-sequence, since we have tir>Vr>Vt, if r > «, and tV > t/, > v„ if r < « ; ^S^K 10 THE PROCESSES OF ANALYSIS. [CHAP. IL and the difference u^ — v^ can be made as small as we please by increasing k. These two sequences u and v therefore uniquely define a real number (rational or irrational) ^, such that { is less than any Dumber in the iz-sequence and greater than any number in the v-sequence, and the differences Ut — ^ and ^—vt can be made as small as we please by increasing k. Then w* - f < w* - v* < ^^^ , so k«*-fl<km-ti*|+|t**-f|<2F:Y + 2^<2^- Moreover, by hypothesis, where p is any positive integer ; and so Since -^j—^ can be made as small as we wish by increasing k, this inequality shews that the sequence a^, aj^, a:,, ... tends to the limit f. Similarly the sequence ^i, ^at yt> ••• tends to a limit rj. Thus if T be any small positive quantity, it is possible to choose a number m such that for all values of r greater than m we have and therefore (xr - f )* + (yr — f y* < t^, or \Zr-l\<T, where Z = f + iff. This inequality shews that the sequence of quantities Zn z^, Zt, .*. tends to the limit I ; which establishes the required result, namely that the condition expressed is sufficient to ensure the existence of a limit. 7. Convergence of cm infinite series. Let til, t^, t^s, ... tin h^ a series of numbers (real or complex). Let the sum be denoted by Sn* Then the infinite series is said to be convergent, or to converge to a sum 8, if the sequence of numbers Si, 82, 89, ... tends to a definite limit 8 aa n tends to infinity. In other cases, the infinite series is said to be divergent. When the series converges the quantity 8 — 8n, which is the sum of the series tifi+i + **fH-« "^ tf n+t + • • • , 7] THE THEORY OF ABSOLUTE CONVERGENCE. 11 is called the remainder after n terms, and is frequently denoted by the symbol Rn. The sum Un+i + Wn+i + ... + thn^ will be denoted by Sn,p- It follows at once, by combining the above definition with the results of the last paragraph, that the necessary and sufficient condition for the convergence of an infinite series is that Sn^p shall tend to the limit zero as n tends to infinity, whatever p is. Since tVM = >S^n,i) it follows as a peurticular case that ibn+i must tend to zero as n tends to infinity, — in other words, the terms of a convergent series must ultimately become indefinitely small. But this last condition, though necessary, is not sufficient in itself to ensure the convergence of the series, as appears from a study of the series In this series, or Sn^n>2' Therefore S = 1 + Si,x 4- iS^., + ^4,4 + fii.« + S^e^ie + ... ^ •■••222 •••> which is clearly infinite ; the series is therefore divergent. Infinite series were used by Lord Brouncker in P/dl, Trans, 1668, and the expressions oonvergent and divergent were introduced by Gregory in the same year. But the great mathematicians of the 18th century used infinite aeries fredy without, for the most part, considering the question of their convergence. Thus Euler gave the sum of the series as zero, on the ground that ...+p + ^ + j+l+«+a«+«»+ (a) ^+^+^+... = -i- (6) 1 ^* and 14. + 4....= ' (c). z sr z— 1 The error of course arises from the fact that the series (6) converges only when | ^ | < 1, and the series (c) converges only when \z\>l,ao the series (a) does not converge for any value otz. The modem theory of convergence may be said to date from the publication of Gauss' DisquigitioneM circa seriem infinitam 1+?^+... in 1812, and Cauchy** Analyse Algdhrique in 1821. See Reifi^ Oeschichte der imendltchen Reihen (Tubingen, 1889). „ .-■ ^.■" ■ W\. . Jl'.' 12 THE PROCESSES OF ANALYSIS. [CHAP. IL 8. AbsoltUe convergence and semi-convergence. In order that the series (which we shall frequently denote by Xun), whose terms are supposed to be any complex quantities, may be convergent, it is sufficient, but not necessary, that the series S | z^ | shall be convergent. For we have I ^n,P I = I ^Wi + ^n+a + ... + ^n+p \ < I l^n+i I + I ^+9 I + ... 4- I Un+p I , and this last expression is inBnitely small, whatever p may be, when n is infinitely great, provided the series S 1 2^ | is convergent. Although this condition is sufficient to ensure the convergence of the series Xun, it is not necessary, i.e. the series Sun can converge even when the series 2 1 Wn | diverges. This may be seen by considering the series 1 2 + 8 4+6 ••• + n +•••• This series is convergent ; for writing it in the form 1.1.1. or 2 12 + 30+*"' we see that its sum is greater than ^ > And that the partial sum obtained by truncating the series after its 2nth term increases as n increases; on the other hand, by writing it in the form we see that the sum is less than 1, and that the partial sum obtained by truncating the series after its (2n + l)th term decreases as n increases. These partial sums must therefore tend to some limit between x and 1, and so the series converges. But the series of moduli is ^2^3^4+'*'' which as already shewn is divergent. In this case therefore, the divergence of the series of moduli does not entail the divergence of the series itself. Series whose convergence is due to the convergence of the series formed by the moduli of their terms possess special properties of great importance, and are called absolutely convergent series. Series which though convergent are not absolutely convergent (i.e. the series themselves converge, but the series of moduli diverge) are said to be semi-convergent or conditionaUy convergent 8, 9] THE THEORY OF ABSOLUTE CONVERGENCE. 18 CO 1 9. The geometrical series, and the series S — . The convergence of a particular series is in most cases investigated, not by the direct consideration of the sum Snp, but (as will appear from the following articles) by a comparison of the given series with some other series which is known to be convergent or divergent. We shall now investigate the convergence of two of the series which are most frequently used as standards for comparison. (1) The geometrical series. The geometrical series is defined to be the series l + irH--j' + ^ + 2^.... Considering the series of moduli we have for it iSfn.i,= kl'^' + l-^^l^+'H- ... +|«^|*^, or ^n.p=l^r^j^. Now if I -2: 1 < 1, then -= — r-K- is finite for all values of jp, while \z\^^^ tends 1-1*. to zero as n tends to infinity. The series 1+ U| + U|>+... is therefore convergent so long as 1 2: { < 1, and therefore the geometric series is ahsolutely convergent so long as \z\<l. When I ir I > 1, the terms of the geometric series do not tend to zero as n increases, and the series is therefore divergent. (2) 2%««ertC8 j;; + 2j + gi+^+g, + .... « 1 Consider now the series 2 — , where s is any positive real quantity. We have 2^ + 3^ < 2"*'^ 2^^ ' 11114 1 4« "^ 5* "^ 6* "^ 7« ^ 4' "^ 4^1 ' and so on. Thus the sum of any number of terms of the series is less than the sum of the corresponding terms of the series 1 _1 11 l»-i "^ 2«-^ "*" 4«"* S*-^ ' 1 J_ 1 1 ^^ p=5"*" 2«-i "^ 2*^"*> ■*"2»(»-i) ■^•••» 14 THE PROCESSES OF ANALYSIS. [OHAP. II. and hence the convergence of this last series would involve that of the original series. But this last series is a geometrical series, and is therefore convergent if 1 , that is, if 8> 1. The series ^ —i is therefore convergent if s>l; and since its terms n=l w* are all real and positive, they are equal to their own moduli, and so the series of moduli of the terms is convergent ; that is, the convergence is absolute. If 5 = 1, the series becomes which we have already shewn to be divergent; and when «»!, it is d fortiori divergent, since the etfect of diminishing « is to increase the terms of the • 1 . series. The series 2 — w therefore divergent if s^l. 10. The Comparison-Theorem. We shall now shew that a series » wUl be absolutely convergent^ provided 1 1^ | w always less than C\vn\, where C is any finite number independent of n, and v^ is the nth term of another series which is known to be absolutely convergent. For we have under these conditions I Un+i i + I Un+t I + . . . + I Un+p I < C^ 1 1 ^n+i | + | Vn+3 I + • • • + | t^n-hp I } > where n and p are any integers. But since the series Svn is absolutely convergent, the series 2 | Vn | is convergent, and so tends to zero as n increases, whatever p may be. It follows therefore that tends to zero as n increases, whatever p may be, i.e. the series 2 | Wn | is convergent. The series 2t*n is therefore absolutely convergent. Corollary, A series will be absolutely convergent if the ratio of its terms, to the corresponding terms of a series which is known to be abso- lutely convergent, is always finite. Example 1. Shew that the series cos « + Si cos 2«H-si cos 3« + 7% COB 4« + . .. 2' o' 4' is absolutely convergent for all real values of t. 10] THE THEORY OF ABSOLUTE CONVERGENCE. 15 For wheD z ia real, we have | cobtl? | ^ 1, and therefore | ^g i ^ "§ • Th® moduli of the terms of the given series are therefore less than, or at most equal to, the corresponding terms of the series 1 11 which by § 9 is absolutely conveigent The given series is therefore absolutely convergent. Example 2. Shew that the series 1 1 V . ••> where z^^(^l+^e^, (n=l, 2, 3, ...) is conveigent for all values of z^ except the values z^z^ z^^z^,*... The geometric representation of comjdez numbers is helpful in discussing a question of this kind. Let values of the complex number z be represented on a plane : then the values 'i) Hi %)••• ^U ^*^i^™ & series of points which for large values of n lie very near the circumference of the circle whose centre is the origin and whose radius is unity: so that in &ct the whole circumference of this circle may be r^arded as composed of points included in the values z^^. For these special values z^ of r, the given series is clearly divergent, since the term becomes infinite when z=z,^. The series is therefore divergent at ^U points z situated on the circumference of the circle of radius unity. Suppose now that z has a value which is distinct from any of the values z^. Then is finite for all values of n, and less than some definite upper limit c : so the moduli of the terms of the given series are less than the corresponding terms of the series which is known to be absolutely convergent. The given series is therefore absolutely convergent for all values of r, except the values z^. It is interesting to notice that the area in the 2-plane over which the series converges is divided into two parts, between which there is no intercommunication, by the circle 1*1-1. Example 3. Shew that the series 2sin^+4sing+8sin — + ... + 2*sin ^-h.-- converges absolutely for all finite values of z. For when n is large, the quantity 2,„ • WW "sm- 2« 3* has a value nearly unity; the given series is therefore absolutely convergent, since the comparison series 2 * ' is absolutely convergent. 16 THE PROCESSES OF ANALYSIS. [CHAP. II. 11. Disctission of a special aeries of importance. The theorem of § 10 enables us to establish the absolute convergence of a series which will be found to be of great importance in the theory of Elliptic Functions. Let Q>i and o), be any constants whose ratio is not purely real; and consider the series Us z^ >r {z — 2ma)i - 2nQ),)* (2ma>i + 2nft)jy where the summation extends over all positive and negative integral and zero values of m and n (the simultaneous zero values m = 0, n = excepted). At each of the points z = 2ma>i + 2nQ>s one term of the series is infinite, and the series therefore is not convergent. The absolute convergence of the series for all other values of z can be established as follows. Let z have any value not included in this set of exceptional values. The series may be written L + s ^ ^ (2ma>i + 2n(o^ 1- r-^- 2mQ>i + 2nci>,> Now when | 2ma>i + ^na>^ \ is large (and we can suppose the series arranged in order of magnitude of | 2m6)i + 2m»2|), we have 1 1 f r_i Limit o = ■!• 2m(it)i + 27ica2 The series is therefore absolutely convergent if the series 2 ^ (2ma>i -♦- 271012)* is absolutely convergent : that is, if the series 2 ^ - (2wicoi -f- 2r2G)a)' is absolutely convergent. To discuss the convergence of the latter series, let cDi = ai + t^Si , ©a = Oa + iySj, where aj, Oa, ySi, ^Sj, are real. Then the series of moduli of the terms of this series is This converges if the series 2 •- (which we may denote by S) (m* 4- n^y ■i 11, 12] THE THEORY OF ABSOLUTE CONVERGENCE. IT converges ; for the quotient of corresponding terms is where M = - ; and this is never zero or infinite. We have therefore only to study the convergence of the series S. Now iit= -00 n= - 00 (m* + n^y 00 00 1 = 4 2 2 where in the summation the occurrence of the pair of values m = 0, n = together is excluded. Separating S into the terms for which m = n, m>n, and m<n, re- spectively, we have 00 1 00 yn-1 1 CO »-l 1 i/Sf=2 -4-i+2 2 --^ — i+2 2 m=i(2m')* w=i n=o (m'4-n*)* »-i »t=o (w' + ?i*)* »»-! 1 Wl 1 But 2 . < i < — ;, . Therefore jfif< | -_^+ f 1+ i 1 t=i 2*m' «=i ^ «=i ^ 00 I ** 1 But the series 2 —1 and 2 -„ are known to be convergent. So the series S is absolutely convergent. The original series is therefore absolutely convergent for all values of z except the specified excluded values. Example, Prove that the series 1 2 {m^-\-m^-\- . . . 4-wir^r in which the summation extends over all positive and n^ative integral values and zero values of m,, m,, ... m,, except the set of simultaneous zero values, is absolutely convergent if yi>- . (Eisenstein, CrelUs Journal^ xxxv.) It 12. A convergency-test which depends on the ratio of the successive terms of a series. We shall now shew that a series ^1 + ^ + Us + ^4 + • • • is absolutely convergent, provided that for all valves of n greater than some w.A. 2 18 THE PROCESSES OF ANALYSIS. [CHAP. II. fixed value r, the quantity \ — ^1 is less than K, where K is some positive quantity Un independent of n and less than unity. For the terms of the series are respectively less than the terms of the series which is a geometric series, and therefore absolutely convergent when K <1. Thus if —^ tends as n increases to a limiting value which is less than unity, the series is absolutely convergent. Example 1. If | c |< 1, shew that the series 11=1 converges absolutely for all values of z. For the ratio of the (n + l)th term to the nth is or c**+V, and if I c |<I, this is ultimately indefinitely smalL Example 2. Shew that the series ,.«"fe^, (a-6)(a-2fe) (a-fe)(a-26)(a-36) converges absolutely so long as 1 2 |<-T-i . For the ratio of the (»+l)th term to the nth is — r ^> ^^ ultimately - 6z : so the con- dition for absolute convei^ence is 1 62; |<1, or U|<--t-: . Example 3. Shew that the series 2 converges absolutely so long as 2|<1. For when |2|<1, the terms of the series bear a finite ratio to those of the series 2 n2"~^; but th^ latter series is then absolutely convergent, since the ratio of the 00 (n+ l)th term to the nth is f 1 H — j z^ which tends to a limit less than unity as n increasea 13. A general theorem on series /or which Limit »=oo ^n+i = 1. It is obvious that if, for all values of n greater than some fixed value r. 13] THE THEOBY OF ABSOLUTE CONVERGENCE. 19 t^n+i I is greater than \iLn\, then the terms of the series do not tend to zero as w« n+i U n n increases, and the series is therefore divergent. On the other hand, if is always leas than some quantity which is itself less than unity, we have shewn in § 12 that the series is absolutely convergent. The limiting case is that in which, as n increases, u n tends to the value unity. In this case a further investigation is necessary. We shall now shew that a series Wi + W2 + t^+ '", in which ^+1 tends to the limit unity as n increases, will he absolutely con- m vergent if, for ail values of n after some fixed value, we have <1- 1 + c n where c is a positive quantity independent of n. For compare the series 2 | t/n | with the convergent series Svn* where A v«= - n i+, and -4 is a constant ; we have Vn Vn+l/ \ nJ -(-I) 1+1 .11 = 1 — -^— -I- terms in — , — , n n^ if V, As n increases, -""^^ will therefore tend to the limit Vn 1- Hi n so that after some value of n we shall have ^+1 Wn Vn+i V n By a suitable choice of the constant A, we can therefore secure that for all values of n we shall have As Sv^ is convergent, 2 1 1^ | is therefore convergent, and so 2wn is abso- lutely convergent. 2—2 20 THE PROCESSES OF ANALYSIS. [chap. n. Corollary. If form ^+1 ^ can be expanded in descending powers of n in the where -4i, A^, -4,, ... are independent of n, then the series is absolutely convergent if -4i < — 1. This is easily seen to follow from the fact that when n is large the terms become unimportant in comparison with A^, 14. Convergence of the hypergeometric series. The theorems which have been given may be illustrated by a discussion of the convergence of the hypergeometric series, a,b a(a + l)b(b+l) ■^l.c "^ 1.2.c(c + l) a(a-H)(a-f 2)6(6-H)(6 + 2) 1.2.3.c(c + l)(c+2) ■*■•••' which is generally denoted by F (a, 6, c, z). If c is a negative integer, all the terms after the (1 — c)th will be infinite ; and if either a or 6 is a negative integer the series will terminate at the (1 — a)th or (1 — 6)th term as the case may be. We shall suppose these cases set aside, so that a, 6, and c are assumed not to be negative integers. The ratio of the (n -♦- l)th term to the nth is Un+i (a -♦- M — 1) (6 4- n - 1) = z. Therefore Un ^n+i w(c+w — 1) U. n 1 + a- 1 n 1 + 6-1 n 1 + c-1 n As n tends to infinity, this tends to the limit \z\. We see therefore by § 12 that the series is absolutely convergent when \z\<l, and divergent when \z\>h When 1^1=1, we have ^+1 1 + g-l n 1 + 6-1 n l_£^^(- !)•_... n n" 1 + - ^ -+ terms m -- , — , etc. n n^ n' Now a, 6, c are in the most general case supposed to be complex numbers. 14, 15] THE THEORY OF ABSOLUTE CONVERGENCE. 21 Let them be given in terms of their real and imaginary parts by the equations a = a' 4- ia!\ c = c' -f ic". Then (neglecting the terms in — , — , etc.) we have 1% Iv ^+1 ajH6W;-H-i(a" + 6"-c") n = 1 + a' + h'-c'-W fa" 4 h" - c"\») * n ) - (" ' „ a' + 6'-c'-l , .1 1 ^ = 1 H h terms m — , — , etc. By § 13, the condition for absolute convergence is a' + 6'~c'<0. Hence when \z\=^\,ihe condition for ike absolute convergence of the hyper- geometric aeries is that the real part of a + b — c shall be negative, 16. Effect of changing the order of the terms in a series. In an ordinary sum the order of the terms is of no importance, and can be varied without affecting^the result of the addition. In an infinite series however this is no longer the case, as will appear from the following example. T^f 'C 1.1 1.1.1 1.1.1 1. and Sf=i -1+1-1+1-1 + and let 2n and Sn denote the sums of their first n terms. These infinite series are formed of the same terms, but the order of the terms is different. Then if A: be any positive integer, n * 11111 But ^*-^*-' = 2/fc— l+2A-]k=2^T-2ifc- Similarly p,_, -p,_ = ^ - -^-_-^ . A series of equations like this can be formed, of which the last is Adding these, we have ,111 1 c» 22 THE PROCESSES OF ANALYSia [CHAP. II. Thus 2jt = So: + s Safc. 2 Making k indefinitely great, this gives an equation which shews that the eflfect of deranging the order of the terms in S has been an alteration in the value of its sum. Example, If in the series the order of the terms be altered, so that the ratio of the number of positive terms to the number of negative terms in S^ is ultimately a\ shew that the sum of the series will become log (2a). (Manning.) 16. The fundamental property of absolutely convergent aeries. We shall now shew that the sum of an absolutely convergent series is not affected by changing in any manner the order in which the terms occur For let iS = t^i + 1^, 4- 1/^ + ^4 + . . . be an absolutely convergent series, and let /S' be a series formed by the same terms in a different order. Suppose that in order to include the first n terms of Sy it is necessary to take m terms of S\ So if k be any number greater than m, we have ^k = 'Sin + terms of S whose suflBx is greater than n. Therefore I 'S'A;' — /S I < I /Sin — /Si j -♦- the sum of the moduli of a number of terms of S whose suffix is greater than n When n tends to infinity, | /?„ — ^ I tends to zero since the series 8 is con- vergent, and the sum tends to zero also, since the series is absolutely convergent. Thus I St' — /Si I tends to zero when k is indefinitely increased; which establishes the required result. 17. Riemann*s theorem on semi-convergent series. We shall now shew that a semi-convergent series Wl + ^ + 1^ + ^4 + . . . , with real terms, may be made to converge to any desired real value, by suitably disposing ike order in which the terms occur. This property stands in sharp contradiction to that proved in the last article; an example of it was afforded by the result of §15. 16, 17] THE THEORY OF ABSOLUTE CONVERGENCE. 28 To establish the theorem, let the positive terms in the series be "^j '^> ^i>,i ••• > and let the negative terms be '^j*,* "^1 "^i*,! ••• • Then the series and -^i»,-^-^ii,- ... cannot be both convergent : for if they were, the original series would be absolutely convergent: one of them must therefore be divergent: and the other cannot be convergent, since in that case the original series would be divergent. It follows that the series tAp, + 1^, + tij^ + . . . and — t^, — ^n,-^— ... are both divergent. Now let S be any real number, and let it be desired to change the order of the terms in the original series, in such a way as to cause it to converge to the sum &. Suppose that a terms of the series have to be taken in order to obtain a sum greater than fif, so that Take now a number 6 of the terms of the series such as are required to make the sum less than & : so that Take next a number c of the terms of the series such as are required to make the sum greater than S ; and then take a number d of the terms of the series in such a way as to make the sum less than & again ; and so on. Proceeding in this way, we obtain a series whose sum at any stage of 24 THE PROCESSES OF ANALYSIS. [OHAP. II. the process, differs from S by less than the last term included. But the terms of the series lii + ^,4- w,+ ... are ultimately indefinitely small, since the series is convergent; we can therefore in this way obtain a series whose sum differs from S by as little as we please ; and it consists of the terms of the original series, disposed in a different order. This establishes the result above stated. Corollary. If the terms of the original series are complex, they can be disposed in such an order as to give an arbitrarily assigned value to either the real or the imaginary part of the sum. 18. Cauchys theorem on the multiplication of absolutely convergent series. We shall now shew that if two series and T= Vi + Vj + Vt 4- ... are absolutely convergent, then the series formed by the products of their termSy written in a/ay order, is ahsohitely con- vergenty and has for sum ST. Suppose that in order to include all the terms of the product it is necessary to take m terms of P ; and let k be any number greater than m. Then -P* = (^ + ^+ ... +^n)(Vi + ^2+ ••• +Vn) + terms t^aV/i in which either a or yS is greater than n, so I Pjt- Sri < I Sr,Tn'-ST\ 4- terms | w. 1 1 v^ |. Let {u-k-p) be the greatest suffix contained in these suffixes a and /8. Then |P,-OT:^|Snrn-«2'| + lhf„+,|4-...4-|^n+pll{iVi;4-...4-|t;n+p|} 4- {i Wi I 4- ... 4- I Mn 1} (i V„+, I 4- ... 4- I Vn+p I}. Now when n tends to infinity, I ^n+i i 4- 1 Un+% j 4- ... 4- 1 Un^p \ tcuds to zero, and I Vn+i I 4- ... 4- 1 Vn^p \ tends to zero, while their coefficients tend to finite limits. Therefore | P^ — ST \ tends to zero, which proves the theorem. 18, 19] THE THEORY OF ABSOLUTE CONVERGENCE. 25 Example 1. Shew that the series obtained by multiplying the two seriee l+£ + ^+f' + ?! + 2 2* 2' 2* ''' and l+- + -2+'5+ — > Z Z /u converges so long as the representative point of z lies in the ring-shaped r^on bounded by the circles l^jal and |z|a2. For the first series converges only when |«|<2, and the second only when |«|>1, and both must converge if the product is to converge. Example 2. Prove by multiplication of series that {cos S? cos 52 1 fTT* 2 /cos 2^ cos 4? . \) . cos a? . cos 5^ . For the coefficient of cos (2r-|-l) 2 in the product on the left-hand side of the equation is _jr« 1 ; 1 f 1 , 1 1 9 (2r+l)« 3 *=, (Uf \{U - 2r- 1)« "^ (2it+2r4- 1)«J ' or or or or or 9(2r+l)2 3(2r+l)«tli\V2it-2r-l 2/?/ "*" W 2Xr+2r+l/ J' ^2 1 • r 2 2 4 11 r_2 2 4 1 t(2i-)2'*"(2it-l)2 (2;fc-2r-l)(2ir4-2r+l)J 9(2r+l)« 3(2r+l)«fcfi l(2i-)2^(2it-l)2 (2;fc-2r- l)(2ir4-2r+l)J ^3(2r4-l)*' 9(2r + l)2'*"3(2r-hl)* 3(2r+l)2V 22'^3«'^42'^"7 '*"3(2r+l)*' »r2 . 1 2 TfS 9(2r+l)2^(2r+l)* 3(2r+l)« ' 6 ' 1 (2r+l)*' which gives the required result. 19. Mertens' theorem on the multiplication of a semi-convergent series hy an absolutely convergent series. We shall now shew that if a series S = i^-fi^ + w, + ... is semi-convergent, and another series is absolutely convergent, then the series where pn = U^Vn + t^Vn-i + ... + UnV^, is convergent, and its sum is ST. For Pn = the sum of all terms UaVp in which a + /S < n + 1 =^(Ui + 1U+,,.+Un)(Vi-{-V2+..,+Vn)-VjUn-Vi{Un'¥Unr^i)-... 26 THB PROCESSES OF ANALYSIS. [CHAP. II. Therefore |Pn-/Sfri<|/Sf„r„-/gr| + |w„||t;,| + |t;,|K+un-,l + ... Now let k denote some number about half-way between 1 and n\ let € be the greatest of the quantities and let 7 be the greatest of the quantities i ^n + . . . + Wn-*-i 1 1 • • • ! 1^ + ^n-i + . . . + ti<| | . Then As n tends to infinity, € and { | y^^., | + . . . + 1 Vn 1 } are infinitesimal, while {|i^s| + ... +1 Vfc+a|} and 7 are finite. So every term on the right-hand side of the last equation is infinitesimal, and therefore in the limit P = /gT, which establishes the theorem. 20. AbeVa resuU on th^ multiplication of aeries. We shall next prove a still more general theorem due to Abel*, which may be stated thus : Let two series 2 u^ and 2 Vn converge to the limits U and V respec- tively, and let the quantity be denoted by Wn. Then if the series converges at all, it converges to the sum UV, It will be noticed that none of the series considered need be absolutely convergent. We shall follow a method of proof due to Cesarof. Lemma I. If a set of quantities «i, 52> *8> ••• t^^ ^ ^ limit s, then . . 1 * Limit - 2 «i = «. f»=oo n ,=1 For if € be any small positive number, we can find a number k such that the inequality \Sr — s\<€ is satisfied for all values of r greater than k. We have therefore 2 n 1 * 1 *"** 1 * - 2 «< = - 2 «i + - 2 « + - 2 («f — 5). Ui^i ni=i n i^jc ni^jc • Crelle'B Journal, i. (1827). t Bulletin des Sciences math. (2) xiv. (1890). 20] THE THEORY OP ABSOLUTE CX)NVERGENCE. 27 Thus - 2 «i s . < - 2 ^< + - 2 5i - 5 1 1 4 1 I n-k + l < - 2 \8i\-\ €. TV t==l W 1 * Now make n infinitely great compared with k ; then - 2 \si\ tends to zero, ^ f=i fl Jfc 4- 1 and tends to unity, 1 \ Limit - 2 «t- - fi < € ; and so and as e can be made as small as we please, this establishes the Lemma. Lemma II, If, as n increases indefinitely^ On and bn tend respectively to the limits a and b, then Limit -{aibn + a^bn-i + . . . -h Onfti) = ab. n=oo n To prove this, let v be the greatest integer contained in ^n. Then if e be any small positive number, we can take n so great that the inequality 6^ - 6 I < 6 holds so long as r > n-^p. Hence |ai(6n-6) + aa(6^i- 6) + ... + a^(6n-r+i — b)\ < €{|ai| + I Oal + ... + I a„|}. Hence Limit - \ Oi (bn— b) + a^(bn^i — b) + ... + a„(6,^_„4.i — 6) | n n=cc <€Limit-{|ai| + | 09! + ... + |a„|} 91=00 n < € I a I , by Lemma I. The right-hand side of this inequality can be made as small as we please; hence Limit - {oi (bn — 6) + a, (b^^i — 6) + ... + a„ (^n-i^+i — b)] = 0, n=<» n or Limit - (ai6„ + a^bn-j -!-...+ a„6»«„+i) n=oo W = ^6 X Limit- (oi + a? + . . . + a^) y=oo ^ = ^a6, by Lemma I. Similarly Limit - (a^+ibn-y + a^ibn-v+j + ... 4- (tnbi) = JaJ. n=« ^ 28 THE PROCESSES OF ANALYSIS. [CHAP. II. Adding the last two equations, we have Limit (Oibn + Ojin-i + . . • + Onbi) = oi, 11=00 which establishes Lemma II. Now let Wn denote the sum of the n first terms of the series considered in the above enunciation of Abel's result, we have where Un and F„ are used to denote the sums of the first n terms of the series U and V, From thiswe have and so by Lemma II. it follows that Limit -(Tri-fTrj+ ... + Trn)= ^F. 1»=ao W But if the set of quantities TTj, TFj, ]F,, ... tend to a limit W, we have by Lemma I. Limit - ( TT, + TT, + . . . + F„) = W. Hence W=UV, which establishes Abel's result. Example 1. Shew that the series 1-2+2-1 + is convergent, but that its square (formed by AbePs rule), 2 V2 is divergent. ■--Hfa-S-a-^)-" Example 2. If the convergent series 0—1 2^ y 4r"^'*' be multiplied by itself, the terms of the product being arranged as iu AbePs result, shew that the resulting series is divergent if r ^ ^, but that it converges to the sum ^S^ when r<i. (Cauchy and Cajori.) 21. Power-Series, A series of the type in which the quantities Oo, ch, a,, a, ... are independent of z, is called a series proceeding according to ascending powers of z, or briefly a power-series. 21] THE THEORY OF ABSOLUTE CONVERGENCE. 29 We shall now shew that if a power-series converges for any value Zq of z, it will he absolutely convergent for all values of z whose representative points are within a cirde^ which passes through z^ and has its centre at the origin. 00 For if z be such a point, we have | -? | < | «o !• Now since ^a^zj^ converges, the quantity a^z^ must tend to zero as n increases indefinitely, and so we can write Jn anl = ^p • • • • where €» tends to zero as n increases. Thus Ittol + loil |^| + |a«|U(^ + ... = €o + €i - Zy Now ultimately every term in the series on the right-hand side is less than the corresponding term in the convergent geometric series z z s Z + e. — + €, Zq Zq ^0 00 n=0 Z ^0 n the series is therefore convergent; and so the power-series is absolutely convergent, as the series of moduli of its terms is a convergent series ; which establishes the result stated. It follows from this that the area in the ^-plane over which a power- series converges must always be a circle ; for if the series converges for any point outside the particular circle which has just been found, we can (by taking this point as the point Zq) obtain a new and larger circle within which the series will converge. The circle in the ^r-plane which includes all the values of z for which the power-series ao + ttiZ + a22^ + a^z^ + ... converges, is called the circle of convergence of the series. The radius of the circle is called the radius of convergence. The radius of convergence of a power-series may be infinitely great; as happens for instance in the case of the series which represents the function sin z ; in this case the series converges for all finite values of z real or complex, ie. over the whole 2:-plane. On the other hand, the radius of convergence of a power-series may be infinitely small ; thus in the case of the series 1 -h 1! ^ + 2! £:«-h 3! -2^ + 4! ^ + ..., we have u n-fi ^n\z\. 30 THE PROCESSES OF ANALYSIS. [CHAP. II. which, for all values of n after some fixed value, is greater than unity when z has any value different from zero. The series converges therefore only at the point -? = 0, and its circle of convergence is infinitely small. A power-series may or may not converge for points which are actually on the circumference of the circle ; thus the series z z^ 2? z* whose radius of convergence is unity, converges or diverges at the point 2: = 1 according as « is greater or not greater than unity, as was seen in § 9. 22. Convergency of series derived from a power-series. Let aQ + aiZ + a^z^ + a^z^-^- a4Z* + ... be a power-series, and consider the series Oi -{• 2aiZ -\' SotZ* + ia^z* -^ ..., which is obtained by differentiating the power-series term by term. We shall now shew that the derived series has the same circle of convergence as the original series. . For let -? be a point within the circle of convergence of the power -series ; and choose a positive quantity r, intermediate in value between | z \ and the radius of convergence. Then, since the series 2 an r^ converges absolutely, its n=0 terms must decrease indefinitely as n increases; and it must therefore be possible to find a positive quantity M, independent of n, such that the inequality M is true for all values of n. Then the terms of the series in\an\\z\^-' are less than the corresponding terms of the series But in this series we have Un n r \ nj r ' which, for all values of n greater than some fixed value, is constantly less than unity ; this comparison-series therefore converges, and so the series in\an\\z\^' 22, 23] THE THEORY OF ABSOLUTE CONVERGENCE. 31 converges ; that is, the series 2 non^^^ converges absolutely for all points z «— 1 00 situated within the circle of convergence of the original series 2 On-s^, and the two series have the same circle of convergence. Similarly it can be shewn that the series 2 -^^^ , which is obtained by integrating the original power-series term by term, has the same circle of 00 convergence as 2 OnZ^K 23. Infinite ProdiLcts. We proceed now to the consideration of another class of analytical ex- pressions, known as infinite products. Let l+Oi, l + Oj, l+Os, ... be an infinite set of quantities. If as n increases indefinitely, the product (H-ai)(l + aj)(l + a,)...(l +an) (which we may denote by 11^) tends to a definite limit other than zero, this is called the value of the infinite product n = (l+ai)(l+a3)(l +0,) ..., and the product is said to be convergent The product is often written n (1 -f- a«). If the value of the product is independent of the order in which the factors occur, the convergence of the product is said to be absolute. The condition for absolute convergence is given by the following theorem : in order that the infinite product (l-f-ai)(l+a,)(l + a,)... may he ahsolutely convei^gent, it is necessary and sufficient that the series ' Oi -f- Oj + aj -h . . . should he ahsolutely convergent. For Iln = e^**'^^^'''^"*"***^"*'*^"^-'*"^**^^"*'*'^, so that n is absolutely convergent or not according as the series log(l + Oi) 4- log (1 + Og) -h log (1+ a,) + ... is absolutely convergent or not. But since log(l + o^) is nearly equal to ar when Or is small, the terms of this series always bear finite ratios to the corresponding terms of the series and so the absolute convergence of one series entails that of the other ; which establishes the result*. * A disooBsion of the convergence of infinite products, in which the results are derived withont making use of the logarithmic function, is given by Pringsheim, Math, Ann. xxxoi. pp. 119—164. ^ 32 THE PROCESSES OF ANALYSIS. [CHAP. II. ExcMiple, Shew that the infinite product sin z sin \z sin ^z sin \z z ' ^z ' '~^z ' iz is absolutely convergent for all values of z, . z sm- For when n is large, ^~~ is of the form 1 — ^ , where \^ is finite ; and the series n * X * 1 2 -^ is absolutely convergent, as is seen on comparing it with 2 ,« . The infinite pro- 11=1 n* n=i w duct is therefore absolutely convergent. 24. Some examples of injmite products. Consider the infinite product m sm 21 which represents the function . In order to find whether it is absolutely convergent, we must consider the * ^2 ^* * 1 series 2 ^-r , or — 2 — ; this series is absolutely convergent, and so the product is absolutely convergent for all finite values of z. But now let this product be written in the form (-i)(>-J)('-i)('-4)-- The absolute convergence of this product depends on that of the series z z z z But this series is only semi-convergent, since its series of moduli z\ \z\ \z\ \z' + — -!--- +'77-'+... 7r TT 27r 27r is divergent. In this form therefore the infinite product is not absolutely convergent, i.e. if the order of the factors ( 1 ± - 1 is deranged there is a risk of altering the value of the product. Lastly, let the same product be written in the form M)'-H('n)«1l('-l^)«"}{('-a- in which each of the expressions 1 + I e mir >■ 24] THE THEORY OF ABSOLUTE CONVERGENCE. 33 is counted as a single term of the infinite product. The absolute convergence of this product depends on that of the series or ( 27r»'*'"V'^( 27r^'^-)"^r27r«.2«'^-'V"^(~27r^:T^'*'--j' and the absolute convergence of this series follows from that of the series The infinite product in this last form is therefore again absolutely convergent, the adjunction of the factors e *'" having changed the con- vergence from conditional to absolute. Example 1. Prove that n -[(1 ) «*[ is absolutely convergent for all values of Zj if c is a constant other than a negative integer. For the infinite product is absolutely convergent provided the series Le. if 2 < — ii^—+ terms m — _, —. etcV is, * 1 and- on comparison with the convergent series 2 —^ , this is seen to be the case. Example 2. Shew that n jl — (l — ) «"*[ converges for all points z situated outside a circle whose centre is the origin and radius unity. For the infinite product is absolutely convergent provided the series n=2 \ nj is absolutely convergent. But as n increases, (1 — ) tends to the finite limit e, so the ratio of the (n + l)th term of the series to the nth term is ultimately - ; there is therefore z absolute convergence when , - <1, or |z|>l. Example 3. Shew that 1.2.3...(w-l) — - -> - — ^- — -- n' z{z+l)iz + 2).„(z-\-n-\) tends to a finite limit as n increases indefinitely, unless « is a negative integer. W. A. 3 34 THE PROCESSES OF ANALYSIS. [CHAP. II. For the expression can be regarded as a product of which the nth term is This product is therefore absolutely convergent, provided the series is absolutely convergent ; and a comparison with the convergent series 2 —^ shews that this is the case. When 2 is a negative integer the expression clearly becomes infinite owing to the vanishing of one of the factors in the denominator. Example 4. Prove that '(■-^)(-i)(-:)('-s)(>-4i)('-i>-'-'--^^ For the given product -■ir:"(-i)(-i)('-;)-{'-<M^)('-K)('-s) *\ ' 2^ 8 4^1 • 2fc-l 2*^*/ := Limit xz (>-;)- ('-s)'*-('-4)-^- (■-«)'- =.Uimte'H^'lH-"'^2hi-k)z(l--^e'(l-{-^e'i(^ since the product whose factors are is ahsolviely convergent and so the order of its factors can be altered. Since log2 = l -Hi-Hi- - > this sliews that the given product is equal to e V wn.z. 26. Cauchys theorem on products which are not absolutely convergent We shall now shew that if ai + aa + cis + a4+ ... is a semi-convergent sei-ies of real terms, then the infinite product (l+OiXl+OaXl + Oj)... 26, 26] THE THEORY OF ABSOLUTE CONVERGENCE. 36 converges (though not absolutely) or diverges {to the value zero), according as the series ai*+ aj'H- 03*+... is convergent or divergent. For the infinite product in question converges (though not absolutely) or diverges (to the value zero) according as the series log (1 + a,) + log (1 + Oa) + ... is semi-convergent or diverges to the value — 00 . n««oo Now since the series 2 On is convergent, the quantities On ultimately «=i diminish indefinitely, and therefore we can write a,» log (1 + On) = a„ - ^ (1 + €n), where |€n| tends to zero as n tends to infinity. »=• If the series 2 a^* diverges, it is clear therefore that the series 2 log(l + a^) n=l must diverge to the value — 00 ; if on the other hand the series 2 an* con- n=l n=sao verges, the series 2 log (1 + On) is convergent. From this the results relating n=l to the infinite product follow at once. 26. Infinite Determinants. Infinite series and infinite products are not by any means the only known cases of infinite processes which can lead to convergent results. The re- searches of Mr G. W. Hill in the Lunar Theory* brought into notice the possibilities of infinite determinants. The actual investigation of the convergence is due not to Hill but to Poincare, Bull, de la Soc. MoUk. de France^ xiv. (1886), p. 87. We shall follow the exposition given by H. von Koch, Ada Math. xvi. (1892), p. 217. Let Aije (i. A: = — X , . . . + 00 ) be a doubly-infinite set of given numbers, and denote by the determinant formed of the quantities ili^ (i, A = — m — . . . + m) ; then if, for indefinitely increasing values of m. the quantity D^ has a determinate limit Z), we shall say that the infinite determinant is convergent and has a value D, In the case in which the limit D does not exist, the determinant in question will be said to be divergent. * Beprinted in Acta Maihematicay rm. pp. 1 — 86 (1886). 3—2 36 THE PROCESSES OF ANALYSIS. [chap. II. The elements Au{i = — <x> ,,. -^ ao) are said to form the prindpcU diagonal of the determinant D ; the elements Aijc{k = oo .,, + oo) are said to form the line i ; and the elements -4ijt(t = — oo...-f-x) are said to form the column k. Any element A^ is called a diagonal or a non-diagonal element, according Sisi = k or i^ k. The element Aq^q is called the origin of the determinant. 27. Convergence of an infinite determinant We shall now shew that an infinite determinant converges, provided the product of the diagonal elements converges absolutely and the sum of th^e non- diagonal elements converges absolutely. For let the diagonal elements of an infinite determinant D be denoted by 1 + aii(i = — X ... + 00 ), and let the non-diagonal elements be denoted by ttifc ( t > A?, , "* J , so that the determinant is . . . X *i- C&_i_i ^—10 ^ 11 ■ * * ... ^0—1 X I ^^'^W ^01 • • • ... ^—1 ^0 ■*■ « ^11 • • • Then since the series S lott i^k^-ao is convergent, the product 00 / 00 p= n (1+ 2 is . 00 \ *•= - 00 is convergent. a* I) Now form the products m = n ( 1 + 2 a^ m / m »„= n (1+ 2 O'ik > then if, in the expansion of P,„, certain terms are replaced by zero and certain other terms have their signs changed, we shall obtain Dm ; thus, to each term in the expansion of Dm there corresponds in the expansion of Pm a term of equal or greater modulus. Now Dm+p — Dm represents the sum of those terms in the determinant Dm+p which vanish when the quantities a{jt {t. A: = + (m + 1) ... ± (m +p)] are replaced by zero ; and to each of these terms there corresponds a term of equal or greater modulus in Pm+p — Pm* Hence I i)m^« — D^ I ^ P«i+ti - Pm- m+p As the quantities Pm, Pm+u ••• tend to a fixed limit, the quantities Dm, Dm+i, •.. will therefore tend to a fixed limit. This establishes the proposition. 27, 28] THE THEORY OF ABSOLUTE CONVERGENCE. 37 28. We shall now shew that a determinant, of the convergent form already considered, remains convergent when the elements of any line are replaced by any set of quantities whose moduli are all less than som£ fixed positive number. Replace, for example, the elements of the line by the quantities which satisfy the inequality I /*r I < M, where fi is a, positive number; and let the new values of 2)t» and D be denoted by D^ and D\ Moreover, denote by Pm and P' the products obtained in suppressing in Pm, and P the factor corresponding to the index zero ; we see that no term of D^ caai have a greater modulus than the cor- responding term in the expansion of /JbPm ] and consequently, reasoning as in the last article, we have which establishes the result stated. Example, Shew that the necessary and sufficient condition for the absolute conver- gence of the infinite determinant 1 oj ... 01 1 oj ... 02 1 O3 ••• is that the series shall be absolutely convergent. (von Koch.) Miscellaneous Examples. 1. Find the range of values of ;s for which the series 28iIl2^-4 8in*2!+8sin«2-...+(-l)* + l2*sin*•2+... is convergent. 2. Shew that the series 1 _ Jl^ _1 1_ is semi-convergent, except for certain exceptional values of z ; but that the series 1+ L+...+ _L_ \ L__ ___L_+_L_+... t z-^-X '" z-\-p^l z-k-p z-^-p-^-l '" z-h2p-^q-\ z+2p'\-q "*' in which (p-^q) negative terms always follow p positive terms, is divergent. (Simon.) 38 THE PROCESSES OF ANALYSIS. [chap. II. 3. Shew that the series 1* 2^ 3* 4^ (l<a<i3) is oouvergent 4. Shew that the series is convergent. 5. Shew that the series (Cesaro.) a+i3«+a3+/3*+... (0<aO<l) (Cesaro.) 7ig*-i 2 ((-3"-'} 2kiw m converges absolutely for all values of «, except the values (a=0, 1; k=0, 1, ...m-1 ; m«=l, 2, ... x). 6. If 8^ denote the sum of the first n terms of a convergent series whose sum is «, shew that 7. In the series whose general term is u^^qn-Pa: ^ , (0<q<l<x) where y denotes the number of figures in the expression of n in the ordinary decimal scale of notation, shew that i_ Limit tin* =y, and that the series is convergent, although the quantity ^^^^^ is infinitely great when n is infinitely great and of the form 1 + lO"- *. (Lerch.) 8. Shew that the series 4 where qn=q'^~n, {0<q<l) is convergent, although the ratio of the (n + l)th term to the nth is greater than unity when n is not a triangular number. 9. Shew that the series (Cesaro.) 00 ^wix where w is real, and where {w-^-ny is understood to mean c*!**^***), the logarithm being taken in its arithmetic sense, is convergent for all values of «, when the imaginary part of X is positive, and is convergent for values of s whose real part is positive, when x is real. • (— l)n+l 10. Shew that the qth power of the convergent series 2 - ;. is convergent when ll=sl w ^ — <r, and divergent when ^ — >r. 9 9 (Cajori.) mSC. EXS.] THE THEORY OF ABSOLUTE CONVERGENCE. 39 11. If the two semi-convergent series i i^y^ and .<^i|^', where r and s lie between and 1, be multiplied together, and the product arranged as in Abel's result, shew that the necessary and sufi&cient condition for the convergence of the resulting series is r+«>l. (Cajori.) 12. Shew that if the series be multiplied by itself any number of times, the terms of the product being arranged as in Abel's result, the resulting series converges. (Cajori.) 13. Shew that the qth power of the series Oj sin ^+02 sin 2^+ . .. +a« sin n^-f • .. is convergent whenever ^ <r, r being the maximum niunber satisfying the relation for all values of n. 14. Shew that if $ is not equal to or a multiple of 27r, and if the quantities Kq, u^, t£„ ... are all of the same sign and continually .diminish in such a way that the limit of tfM is zero when n is infinite, then the series Stt^ cos {n6+a) is convergent Shew also that, if the limit of t^ is not zero, but all the other conditions above are B , 6 . . satisfied, the sum of the series is oscillatory if - is commensurable, but that, if - is in- commensurable, the siun may have any value between certain limits whose difference is a cosec^^ where a is the limit of u^^ when n is infinite. (Cambridge Mathematical Tripos, 1896, Part I.) 15. Prove that i{('-0 .*-i s> + »*-^^+.+ ^1 }■ where k is any positive integer, converges absolutely for all finite complex values of z. 16. Let 2 ^M be an absolutely convergent series. Shew that the infinite determinant A(C) = (c-4)»-(?o -<?, -6, 4?-e, (c-2)»-<»o 2»-<?o' -61 2»-tf, 4»-(9. 4»-<?<, 2«-tf« -0. 4«-tfo 0»-A -6y (c+2)»-Oo 4»-A 4«-(?„ -^4 2»-tfo (c + 4)»-^o converges : and shew that the equation ia equivalent to the equation A(c)-0 sin' Jwc — A (0) sin* ^ir^o • (HilL) CHAPTER III. The Fundamental Properties of Analytic Functions ; Taylor's, Laurent's, and Liouville's Theorems. 29. The dependence of one complex number on another. The problems with which Analysis is mainly occupied relate to the dependence of one complex number on another. If z and f are two complex numbers, so connected that the value of one of them is determined by the value of the other, e.g. if f is the square of z, then the two numbers are said to depend on each other. This dependence must not be confused with the most important case of it, which will be explained later under the title of analytic functionality. If ^ is a real function of a real variable z^ then the relation between ( and z, which may be written f=/(«), can be visualised by a curve in a plane, namely the locus of a point whose coordinates referred to rectangular axes in the plane are (z, C), No such simple and convenient geometrical figure can be found for the purpose of visualising an equation considered as defining the dependence of one complex number f=f +ii; on another complex number z=x-\-iy. A representation strictly analogous to the one already given for real variables would require four-dimensional space, since the niunber of quantities f > »7i ^1 y? is now four. One suggestion (made by Lie and Weierstrass) is to use a doubly-manifold systeln of lines in the quadruply-manifold totality of lines in three-dimensional space. Another suggestion is to represent £ and 17 separately by means of surfaces A third suggestion, due to Heffter*, is to write then draw the surface r^r{x,y) — which may be called the modular-surface of the function — and on it to express the values of B by surface-markings. It might be possible to modify this suggestion in various ways by representing B by curves drawn on the surface r=^r{x, y). * ZeitschriftfUr Math. u. Phys. xliv. (1899), p. 236. 29, 30] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 41 30. Continuity, Let f(z) be a quantity which, for all values of z lying within given limits, depends on z. Let Zi be a point situated within these limits. Then f(z) is said to be continuous at the point -^i, if, corresponding to any given positive quantity €, however small, a finite positive quantity 17 can be found, such that the inequality i/w-/(*,)i<« is satisfied so long as | ^ — ^1 { is less than 1;. U f(z) is continuous at z = Zi, and if its real and imaginary parts be denoted by u and v, then u and v depend continuously on z. For if f(z) = w + iv, we have I (u - 1^) + i (v - Vi) I < 6, and so {u — v^y -f (v — v^Y < €*, which gives {u — v^y < e^ and {v — ViY < e^, and so | u — -Mj | < e and \v — Vi\<€. The popular idea of continuity, so far aa it relates to a real variable ( depepding on another real variable z, is somewhat different to that just considered, and may perhaps best be expressed by the definition " The quantity f is said to depend continuously on z H as z passes through the series of all values intermediate between any two adjacent values Zi and z^, ( p&sses through the series of all values intermediate between the corresponding values d and fg-" The question thus arises, how far this popular definition is equivalent to the analytical definition given above. Cauchy shewed that if a real variable (, depending on a real quantity «, satisfies the analytical definition, then it also satisfies what we have called the popular definition. But the converse of this is not true, as was shewn by Darboux. This fact may be illus- trated by the following example*. Let B(^) denote the integer next less than x ; and let f(s)^.[l-E |^-^-^}]+^{j^} si IT sm^^ At a;=0, we have/(^)=»0. Between 4?= - 1 and 47= + 1 (except at a;=0), we have /(;r)=sin£. From this it is easily seen that/(^) depends continuously on x near a;=:0, in the sense of the popular definition, but is not continuous in the sense of the analytical definition. * Dae to MansioD, Mathens^ ix. (1899). 42 THE PROCESSES OF ANALYSIS. [CHAP. III. 31. Definite integrals. Let Zo and Z be any two values of z ; and let their representative points A and B in the ^-plane be connected by an arc (straight or curved) AB; and let Zi, z^, Zg, ... Zn be a number of points taken on the line AB in any manner. Let f(z) be a quantity which, for variations of z along the arc AB, depends continuously on z. Let Zq be any point situated in the interval ZqZi of the curve : let Zi be / any point situated in the^||ntervap^i2:,: and so on: and consider the sum S ^f{z^){z, - Z,)'^f{z^){z, - -^0 + ... +/(0(^ - ^n). We shall shew that if the number n increases indefinitely^ in such a way that each of the quantities \ z^ — -?r-i I tenis to zero, then this sum wiU tend to afia>ed limit, independently of the way in which the points Z\i Z^y ••• Zn, Zq , Zi f ... Zn t are chosen. For let € be a given small positive quantity. Since f{z) is continuous, for each point z=^a of the arc AB we can find a quantity 7}a such that \f(z) -/(a) I < e. 80 long as \z — a\<rfa' Let 7} be the least value of rja corresponding to points a on the arc AB. We shall suppose the subdivision of the arc has been carried so far that each quantity Ur — '^r-i| is less than tf, and shall first find the effect of putting in further subdivisions. Suppose then that the interval ZqZi is subdivided at points z^, z^, "*z^^\ that the interval z^z^ is subdivided at the points ^h> '^i9» ••• Zir^\ and so on : so that the sum s becomes «' =f(Zo")(Zoi - Zo) -^f(Zoi)(Zoi - ^oi) + . . . +/(0(^ii - ^i) +f{Zu){z,^ - ^ii) + . . . i" • • • » where z^' is any point in the interval z^z^, z^( is any point in the interval ZnZoi, and so on. Then s-s^ {f(zn ^f(Zo')] (^01 - ^o) + {f(zoO -/(V)! (^« - -^0,) + . . . + l/(^i") -fM] (^n - Z,) + {/(^„') -/(^/)j {z,,^Zu) + ... ^r • • . . 31] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 48 Therefore |«'-«| <€ {|^W-^o| + |'2oJ-'^Ol|+ •••} < € X the length of the broken line connecting the points -e^o» % i -^^w* • • • where I is the length of the arc AB, Now by making € indefinitely small, we can make the right-hand side of this equation as small as we please; and therefore the sum s tends to a definite limit when the number of subdivisions is indefinitely increased, provided that at each change in the subdivisions the old points of division are retained. The restriction contained in the last phrase has still to be removed. To do this, suppose that two different methods of division, in eaxih of which the quantities \Zr — Zr^i\ are less thfin 17, furnish sums s^ and «9. Now combine the two methods of division, so that every point of division in either of the original sjchemes becomes a point of division in the new scheme. Let the sum corresponding to this new method of division be Sj^, Then since by the above I «i — «ia I < €/ and I «s — «u I < ely we have I «i — *a I < 2€Z, which shews that «i and $2 tend to the same limit. The theorem is thus established. The limit thus shewn to exist is called the definite integral of f{z), taken along the arc AB\ it is denoted by I /: f{z)dz^ AB in cases where there is no ambiguity as to path, it may be denoted by f(z) dz. As an example* of the evaluation of a definite int^;ral directly from the definition, suppose it is required to find the definite integral of the continuously dependent quantity (1 -«*)"*, taken along the straight line (part of the real axis) joining the origin (««0) to a point z=Z^ where Z is real Denote the definite int^;ral by /. Then by definition, /= Limit 2 ^^'"^^ and the mode of choosing the points z^ and V is arbitrary, within the limits already explained ; we shall take «,i=sinrd, V = sin(r+i)d, where d= r sin"* Z. n+l • Netto, Zeitschrift/Ur Math. xl. (1896). 44 THE PROCESSES OF ANALYSIS. [CHAP. III. Thus /-Limit I ^^^(r+l)b-^smrb n ^ = Limit 2 ^ sin 3 = Limit 2 (n+ 1) sin 5 . b sm- = sin ~^^ Limit 1 2 =8in-iZ. The value of the definite integral is therefore sin"^ Z, 32. Limit to the value of a definite integral. Let M be the greatest value of !/(^)| at points on the arc of inte- gration AB, Then |/(V) {zi - z,) -\-f{z^) {z, - ^,) + • • • +/( V) {Z - Zn) \ <|/(V)l|^i--^o| + |/(Olk2--^i|+...+|/(V)|l^-^n| ^ ilf {I ^1 - £-0 I -h i ^, - -?i I -h ... + I ^- -^n '} where I is the length of the arc of integration AB. We see therefore, on proceeding to the limit, that f{z)dz AB cannot be greater than the quantity ML 33. Property of the elementary functions. The reader will be already familiar with the word function^ as used (in text-books on Algebra, Trigonometry, and the Differential Calculus) to denote analytical expressions depending on a variable z ; such for example as z^, c*, \ogz, svxr^z^. These quantities, formed by combinations of the elementary functions of analysis, have in common a remarkable property, which will now be investi- gated. Take as an example the function e^. Write e* =f{z). Then if z' be a point near the point z, we have z — z z — z z - z I ( z-/ (z-zY ) 32 — 35] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 45 and benoe, if the point z tends to coincide with z, the limiting value of the quotient z' -z is (F. This shews that ^e limiting value of f{z'y^f{z) z'^z %8 in this case independent of the direction of the short path by which the point / 7noves towards coincidence with z, i.e. it is independent of the direction in which / lies as viewed from z. It will be found that this property is shared by all the well-known elementary functions ; namely, that if f{z) be one of these functions and h be any small complex quantity, the limiting value of Jl/(^ + A)-/(^)} is independent of the mode in which h tends to zero. 34. Occasional failure of the property. For each of the elementary functions, however, there will be certain points z at which this property will cease to hold good. Thus it does not hold for the function at the point z = a, since the limiting value of z ^ a If 1^ 1^ h z — a — h z — a] is not finite when z=^a. Similarly it does not hold for the functions log^ and z^ at the point z = 0. These exceptional points are called singular points or singularities of the function f(z) under consideration ; at other points the function is said to be regular, 36. The analytical function. The property noted in § 33 will be taken as the basis of our definition of an analytic function, which may be stated as follows. Let an area in the -e-plane be given ; and let w be a quantity which has a definite finite value corresponding to every point z in that area. Let z, z-^-izhe values of the variable z at two neighbouring points, and UyU-\rhu the corresponding values of u. Then if at every point z within the area ^ tends to a finite limiting value when hz tends to zero, independently of 46 THE PROCESSES OF ANALYSIS. [CHAP. Uh the way in which Bz tends to zero, u is said to be an analytic function of z, regular within the area. We shall generally use the word " function " alone to denote an analytic function, as the functions studied in this work will be almost exclusively analytic functions. In the foregoing definition, the function u has been defined only within a certain area in the ^-plane. As will be seen subsequently, however, the function u will generally exist for other values of z not excluded in this area : and (as in the case of the elementary functions already discussed) may have Angularities, for which the fundamental property no longer holds, at certain points outside the limits of the area. The definition of functionality must now be translated into analytical language. If /(^) be a function of z, regular in the neighbourhood of a particular value Zy then, by the definition, the quantity z — ^ tends to a definite limit, depending only on z, when / tends to z. Let this limit be denoted by the symbol /' (z). Then (by the definition of a limit) for every positive quantity 6, however small, it is possible to find a quantity 17, such that is less than €, so long as \z — z\ is less than 17. If therefore we write /(/) =f{z) + (/ - Z)f' {Z) + 6 (/ - Z\ we see that | e' | is less than e, so long as | / — ^ | is less than 17 ; that is, the function /(-gr) must be such that the quantity e', defined by the equation /(/) =/(^) + (/ - z)f (^) + « (/ - z\ tends to the limit zero as z tends to z. The necessity for a strict definition of the term "function" may be seen from the following consideration. Let y denote the temperature at a certain place at time U As t varies, y will vary, and y may loosely be called a "function" of t But y cannot be expressed in terms of t by a Maclaurin's infinite series '-w.-.-^'(l),./^(S), I • • • > ** \w*V<-0 for if it could, the knowledge of the temperature for a single day would enable us to determine the quantities 36j THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 47 and then from the Maclaurin's expansion it would be possible to predict the temperature for the future ! Maclaurin's series is in fact, as will appear subsequently, applicable only to analytic functions, in the sense in which analytic functions have been defined above. 36. Cauchys theorem on the integral of a function round a contour. A simple closed curve in the plane of the variable z is often called a contour : if -4, jB, C, D be points taken in order along the arc of the contour, and i{f(z) be a quantity depending on z and continuous at all points on the arc, then the integral / f(z)dz, ABCDA taken round the contour, starting from the point A and returning to A again, is called the integral of the quantity f(z) taken round the contour. Clearly the value of the integral taken round the contour is unaltered if some point in the contour other than A is taken as the starting-point. We shall now prove a result due to Cauchy, which may be stated as foUowa If f(z)i8 an analytic function, regular at all points in the interior of a contour, then I Az)dz=0, where the integration is taken round the contour. For let A, B, (7, D be points in order on the contour. Join A to C hy an arc AECy which will divide the region contained within the contour into two distinct portions. Then the integral taken round the contour ABCDA is equal to the sum of the integrals taken round the two contours ABCEA and ABCDA ; for f f{z)dz+{ f{z)dz J ABCEA J ABCDA = f f{^)dz+j f(z)dz+f f(z)dz-^j f(z)dz J ABC J CEA J ABC J CD A = f f{z) dz. J ABCDA since the integrals along CEA'^ and AEC neutralise each other. Now join any point E on the arc AEC to D by an arc EFD, and join ^ to jS by an arc EOB; then in the same way we see that the integral round ABCEA is equal to the sum of the integrals round ABOEA and EOBCE, and the integral round AECDA is equal to the sum of the integrals round AEFDA and DFECD. Thus the original contour-integral is equal to the sum of the integrals 48 THE PROCESSES OF ANALYSIS. [CHAP. III. round the four contours ABGEA, EGBCE, AEFDA, DFECD, into which it has been divided by drawing the cross-lines. Proceeding in this way by drawing more cros^-lines, we see that the original contour-integral can be decomposed into the sum of any number of integrals round smaller contours, which constitute a network filling up the original contour. Now suppose that each of these small contours has linear dimensions of the same order of magnitude as a small quantity l. Let z^^ be a point within one of them. Then on this small contour we have f{z) ^f(Zo) + (Z- Zo)f (Z,) + (^ - Zo) €, where e is infinitely small when I is infinitely small. Thus ^f{z)dz^^f{z,) dz-^-jiz - Zo)f{z,) dz+j{z--Zo)€dz, where all the integrals are taken round the small contour. Now j / (zo) dz =f(zo) j dz =/(zq) X the increase in value of z after once* describing the small contour = 0. Similarly j/(zo) (z - z^) dz = lf(Zo)jd {(z - Zof] = 0, when the integral is taken round the small contour. Thus, if 7} be the greatest value of | e | for points on the small contour, we have jf(z)dz^^rfl\z-Zo\\dz\, where the integrals are taken round the small contour. Now the right-hand side of this equation is clearly of the order rjl^ of small quantities. The value of jf(z)dz, taken round the small contour, is there- fore a small quantity of order tflK Now the number of such small contours ip a given area is of the order 1 . ' T^. If 77' be the maximum value of 7} for all the small contours in the area, we see therefore that the total sum of the integrals for all the small contours in the area is at most of the order r)'l^ Xj^ or 1;'; and 17' can be made indefinitely small by decreasing I. It follows, therefore, that the sum of the integrals round all the small 36] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 49 contours is zero; that is, the integral round the original contour is zero, which establishes Cauchy's result. Corollary 1. If there are two paths ZqAZ and ZoBZ from Zq to Z, and if f(z) is a regular function of z at all points in the area enclosed by these two rz paths, then / f(z)dz has the same value whether the path of integration is z^AZ or ZqBZ. This follows from the fact that ZqAZEzq is a simple contour, and so the integral taken round it (which is the diflference of the integrals along z^AZ and z^BZ) is zero. Thus, i{f{z) be an analytic function of z, the value of I f(z) dz is to a certain extent independent of the choice of the arc AB, and depends only on the terminal points A and B. It must be borne in mind that this is only the case when f{z) is an analytical function in the sense of § 35. Corollary 2. Suppose that two simple closed curves C^ and Ci are given, such that Cq completely encloses Cy, as e.g. would be the case if Ci and Ci were coucentric circles or confocal ellipses. Suppose moreover that f{z) is an analytic function, which is regular at all points in the ring-shaped space contained between C^ and (7i. Then by drawing a network of intersecting lines in this ring-shaped space, we can shew exactly as in the theorem just proved that the integral ff(z)dz is zero, where the integration is taken romfid the whole boundary of the ring- shaped space; this boundary consisting of two curves Co and Ci, the one described in a positive (counter-clockwise) direction and the other described in a negative {clockwise) direction. Corollary S. And in general if any connected region be given in the ^- plane, bounded by any number of curves C?o, Ci, C„ ..., and if f(z) be any function of z which is regular everywhere in this region, then j/(z) dz is zero, where the integral is taken round the whole boundary of the region; this boundary consisting of the curves Cq, Ci, ... , each described in such a sense that the region is kept either always on the right or always on the left of a person walking in the sense in question round the boundary. An extension of Cauchy's theorem I f{z)dz=0, to curves lying on a cone whose vertex is at the origin, has been made by Raout {Nouv. Annales de Math. (3) xvi. (1897), w A. 4 50 THE PROCESSES OF ANALYSIS. [CHAP. IIL pp. 365-7). Osgood {BvU. Amer, Math, Soc,, 1896) has shewn that the property jf{z) dz—0 may be taken as the defining-property of an analytic function, the other properties being deducible from it. Example. A ring-shaped region is bounded by the two circles U| = l and U|=2 in the /dz — , where the integral is taken round the boundary of this r^on, is zero. For the boundary consists of the circumference |«| = 1, described in the clockwise direction, together with the circumference !«| = 2, described in the counter-clockwise direction. Thus if for points on the first circumference we write «=«**, and for points on the second circumference we write z=2e^, then B and (j) are real, and the integral becomes -«» t.^de . f^ i.2e^d4> 2e^ Jo ^' ^Ji or -2frt + 2fri, i.e. zero. 37. The vaiiie of a function at a pointy expressed as an integral taken round a contour enclosing the point. Let (7 be a contour within which f(z) is a regular function of z. Then if a be any point within the contour, the expression z— a represents a function of z, which is regular at all points within the contour C except the point z^a, where it has a singularity. Now with the point z^sa as centre, describe a circle 7 of very small radius. Then in the ring-shaped space between 7 and C, the function /(f) z — a is regular, and so by Corollary 2 of the preceding article we have f fiz)dz r f(s)dz ^^^ J c Z'-a J y z — a where I and I denote integrals taken in the positive or counter-clockwise sense round the curves C and 7 respectively. But (§35) f/(fM^,f/(«)-K^-")/'(«) + '(^-«)rf, ^ Jy z-a Jy z—a ' 37, 38] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 51 where € is a quantity which tends to zero when the radius of the circle 7 is indefinitely diminished. Thus Jcz-a -^^^lyZ-a Jy Jy Now if at points on the circumference 7 we write z — a- re^, where r is the radius of the circle 7, we have f dz P^ir^^de . r^' ,^ ^ . and f dz=^jire^d0 = O; I f I also I edzl^rj. iirr, where 17 is the greatest value of |€| for points ^ on 7; and therefore in the limit when r is made indefinitely small we have /. edz = 0. y "- L4^-2"/<»). C or •^ ^ ^ 27n J c z — VL This remarkable result expresses the value of a function /(^) at any point a within a contour (7, in terms of an integral which depends only on the value of f(z) at points on the contour itself. Corollary, If f(z) is a regular function of ^ in a ring-shaped region bounded by two curves C and C, and a is a point in the region, then -^ ^ "^ zmjoz — a imjcz — a where C is the outer of the curves and the integrals are taken in the positive or counter-clockwise sense. 38. The Higher Derivates. The quantity /'(-gr), which represents the limiting value of f{z + h)-f{z) h when h tends to zero, is called the derivaie o{ /(z). We shall now shew that /' (z) is itself an analytic funcHon of z, and consequently Usdf possesses a derivate. 4—2 52 THE PROCESSES OF ANALYSIS. [CHAP. HI. For if (7 be a contour surrounding the point z, and situated entirely within the region in which f{z) is regular, we have *=o 27riA \j c Z'-a — h j c z — a \ = J^r/M^+LimitA[ /(^) ^^ Now f /(^)^^ J c(z — ay{z — a-'h) is a finite quantity, since the integrand (z-ay{z-a-h) is finite at all points of the contour (7, and the path of integration is of finite length. Hence Limit ^-; <^ rT = 0» and consequently /' (a) = ^ j ^^^ > a formula which expresses the value of the derivate of a function at a point as an integral taken round a contour enclosing the point. From this formula we have, if h be any small quantity, f (a + h) -f (g) 1 [ mdz \ 1 1_) h 2m} c h \{z-a-hy {z-af) 2mJc {z-a-hy(z-ay zmj c {z — o) where ^ is a quantity which is easily seen to remain finite as h tends to zero. Therefore as h tends to zero, the expression /'(a+h)-f'ia) h tends to a limiting value, namely 2 /■ /{z) dz 2'iTiJc{z— ay ' 38] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 63 The quantity /' (a) is therefore an analytical function of a ; its derivate, which is represented by the expression just given, is denoted by /" (a), and is called the second derivate of /(a). Similarly it can be sheAvn that f" {a) is an analytical function of a, possessing a derivate equal to 3 r f{z) dz ^ iJciz-a/' this is denoted by /'" (a), and is called the third derivate of /(a). And in general an nth derivate of/ (a) exists, expressible by the integral n\ f f (z) dz and having a derivate of the form (n+1)! l)!r f{z)dz , 27rt ic{z-ay^' this can be proved by induction in the following way. Then /(»)(a + A)-/(») (a) n\ [ J(z)dz ( 1 1 ] h i-rnio h l(«-a-A)»+' (^-a)*+'j ^ n!_f f{z)dz U h \-»->_,) 2iriJc(^-a)"*'AlV e-a) ) (n + l)! f f{z)dz + terms which vanish when h tends to zero. which establishes the required result. A function which possesses a first derivate at all points of a region in the ^-plane therefore possesses derivates of all orders. Exam'ple 1. Verify the theorem by use of Taylor's Theorem. By Taylor's Theorem we have 54 THE PROCESSES OF ANALYSIS. [CHAP. III. f dz But when i: is an integer other than unity, I ^ _ ^ is zero, since resumes its original value after describing the contour. So the only surviving part of the right- hand side is -zr- ./H (a) / , or/(*) (a). Example 2. Verify the same theorem by means of integration by parts. We have nl[ f{z)dz _ ( {n-\)\ f{z) \ (n-iy. f r{z)dz f(z) and the first term is zero, since 7-- ^. resumes its original value when z makes the circuit (z—ap of the contour C. Proceeding in this way, we have n! f f(z)dz ^ 1 f f!!Hl)dz 39. Taylor's Theorem. CoDsider now a function f{z\ which is regular in the neighbourhood of a point z^a. Let C be the circle of largest radius which can be drawn with a as centre in the ^-plane, so as not to include any singular point of the function f{z)\ so that f{z) is a regular function at all points of G. Let 5sa + A be any point within the circle G. Then by §37, we have ^/ IN If f{e)dz •' ^ ^ zm Jc z — CL — h But at points z on the circle G. the modulus of ^ , will not exceed ^ z—a—h some finite quantity M, Therefore • 1 r /{zydz.h""^^ M.^irR /[A_|\«+i 27riJa(^-a)«+^(^-a-A) ^ 2ir \RJ ' where R is the radius of the circle G, so that 27ri2 is the length of the path of integration in the last integral, and R^\z — a\ for points z on the cir- cumference of G, The right-hand side of the last inequality tends to zero as n increases indefinitely. We have therefore 39] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 56 which we can write « This- result is known as Taylor's Theorem] the proof we have given is due to Cauchy, and shews exactly for what range of values of z the theorem holds true, namely for all points z which are nearer to a than the nearest singularity of/ {z). It follows that the radios of convergence of a power-aeriee %8 always such as jtist to exclude from, the circle of convergence the nearest singularity of the function represented by the series. At this stage we may introduce some terms which will be frequently used. If/(a) = 0, the function /(-^) is said to have a zero at the point -^ = a. If at such a point/' (a) is diflferent from zero, the zero of /(a) is said to be simple; if, on the other hand, the quantities /'(a), /''(a), .../<**~^* (a) are all zero, so that the Taylor's expansion of f(z) at ^ = a begins with a term in {z — ay*, then the function f(z) is said to have a zero of the nth order at the point z=^a. Example 1. Find a functiou /(«), which is regular within the circle C of centre at the origin and radius unity, and has the value a-oos^ sin^ a«-2aco6^+l a*-2acos^+l (where a>\ and B is the vectorial angle) at points on the circumference of C, We have f{z) dz c ^•^^ « **• . tad . —3 — r :r-— , puttmg «■■«•* ^j, since the only non-zero term is that from k^n. a^ Therefore by Maclaurin's Theorem*, or /(«)«- • for all points within the circle. This example raises the interesting question, What isf(z) for points outside the circle? Is it still — ? This will be discussed m §§ 41, 42. «0 Example 2. Prove that the arithmetic mean of all values of «"* I a^«f, for points « on the circumference of the circle |£|«1, is a«, if Sa^^*" is regular at all points within the circle. ♦ The result /W=/(0) + */'(0)+^/"(0)+.... 2 which is obtained by putting asO in Taylor's Theorem, is osoally called Maclaurin*i Theorem, 56 THE PROCESSES OF ANALYSIS. [CHAP. III. • f{v) (0) Let 2 avz^=if{z)y so that a^^ — \- . Then the required mean is — • I —^^Tr- J where G is the circle, or 27rt, , /(H) (0) or J \ I ni or a,j. f » Example 3. Prove that if A is a given constant, and (1 - 2zh+h^)~^ is expanded in the form l+hP^(z)+h^Pi{z)-{-¥P^(z) + (A), where F^ {z) is easily seen to be a polynomial of degree n in Zy then this series converges so long as ; is in the interior of an ellipse whose foci are the points s=l and z^-l, and whose semi-major axis is 5 (A+ j) . Let the series be first regarded as a function of h. It is a power-series in A, and therefore converges so long as the point h lies within a circle on the A-plane. The centre of this circle is the point A=0, and its circumference will be such as to pass through that singularity of (1 -2aA+ A*)"* which is nearest to A=0. But l-2zh+h^=(h'Z+^^^) (A-«- V^^-^l), so the singularities of (1-2M4-A*)"* are the points A=«-(«*-l)* and A««+(«*-l)*, at which it is infinite. Thus the series (A) converges so long as |A| is less than either U-(2«-l)*| or !«+(«»- 1)*|. Now draw an ellipse in the z-plane passing through the point z and having its foci at the points 1 and - 1. Let a be its semi-major axis, and 6 the eccentric angle of z on it. Then z=acoB6+i{a*-'l)^Bin6y which gives z±{s^- l)*={a±(a*- 1)*} (cos d+t sin 6\ • so |«±(22-l)*|-a±(a8-l)*. Thus the series (A) converges so long as A is lees than the least of the quantities a'^{a^- 1)* and a-{a^— 1)*, i.e. so long as A is less than a—ia^- 1)*. But A=a— (a*-l)* when <*=3(^+i)- Therefore the series (A) converges so long as ^ is within an ellipse whose foci are I and - 1, and whose semi-major axis is - I^+t ) • 40. Forma of the remainder in Taylors Series, The form found in the last article for the remainder after n terms in Taylor's series is f(z) JC^dz 40, 41] THE FUNDAMENTAL PBOPERTIES OF ANALYTIC FUNCmONS. 57 It is not difficult to derive from this expression the forms of the remainder usually given in treatises on the Diflferential and Integral Calculus. For on mtegrating by parts the quantity n I ^>— — _ .x^-h , we have Jo (z-a- 1)"^^ {z - a)«+^ ^Kn-i-^) ]^ ^^ ^a^ty^^ k^ h^ 1 "• / - " _ W4.« •"•••> {z - a)»*+i {z - a)'»+« by successive repetition of this process, A« ' "" (2r — aY{z -a-hy which is a new form for the remainder. Now suppose that all the quantities concerned are real. Then along the line of integration, {h — ^)**~^ has a fixed sign, so ^»=(^?i)i/„*<'^-'>"-''^' where H lies between the greatest and least values of/<**>(a + ^) between < = and t = k We can therefore write H =/<*»> (a + Oh), where < d < 1, and then or iJ = ^/(»)(a + ^A), which is Lagrange's form for the remainder, Darboux gave in 1876 {Journal de Math. (3) ii. p. 291) a form for the remainder in Taylor's Series, which is applicable to complex variables and resembles the above form given by Lagrange for the case of real variable& 41. The Process of Continuation, Near every point P{z^ at which a function f{z) is regular, we have seen that there is an expansion for the function as a series of ascending positive integral powers of {z — z^, the coefficients in which are the suc- cessive derivates of the function at z^. Now let A be the singularity o{ f(z) which is nearest to P, Then the circle within which thus expansion is valid has P for centre and PA for radius. 58 THE PROCESSES OF ANALYSIS. [CHAP. in. Suppose that we are given the values of the function at all points of the circumference of this circle, or more strictly speaking, of a circle slightly smaller than this and concentric with it : then the preceding theorems enable us to find its value at all points within the circle. The question arises, How can the values of the function at points ouUide the circle be found ? In other words, given a power-series which converges and represents a fwnction only at points within a circle, to derive from it the values of the function at points outside the circle. For this purpose choose any point Pi within the circle, not on the line PA, We know the value of the function and all its derivates at P,, from the series, and so we can form the Taylor series with Pi as origin, which will represent the function for all points within some circle of centre P,. Now this circle will extend as far as the singularity which is nearest to Pi, which may or not be A ; but in either case, this new circle will generally* lie partly outside the old circle of convergence, and for points in the region which is included in the new circle hut not in the old circle, the new series tvill furnish the values of the function, although the old series failed to do so. Similarly we can take any other point P^, in the region for which the values of the function are now known, and form the Taylor series with P, as origin, which will in general furnish the values of the function for other points at which its values were not previously known ; and so on. This method is called continuation'^. By means of it, starting from a representation of a function by any one power-series we can find any number of other power-series, which between them furnish the value of the function at all points where it exists ; and the aggregate of all the power-series thus obtained constitutes the analytical expression of the function. Example, The aeries represents the function 1 £ £« «3 ' ^ ' a-z only for points z within the circle \z\=a. But any number of other power-series exist, of the type 1 , r~6 (z-by (z-bf a-6"^(a-6)2'*"(a-6)s'*"(a-6)*"*"'*'* which represent the function for points outside this circle. * The word "gfenerallj" must be taken as referring to the cases which are likely to come under the student's notice before he reads the more adyanoed parts of the subject, t In Qerman, Fortsetzung. 42] THB FUNDAMENTAL PROPERTIES OF ANALYTIC FCTNCTIONS. 59 On functions to which the continuation-process cannot be applied. It is not always possible to carry out the process of continuation. Take as an example the function f{z) defined by the power-series /(«)=1 +««+«♦ -f^^+^W-H.. .+««" + ..., which clearly converges in the interior of a circle whose radius is unity and whose centre is at the origin. Now as t approaches the value +1 by real values, the value of /(«) obviously tends towards -foo ; the point +1 is therefore a singularity of f{z). But /W=^*+/W, 80 if « is such that -?*=1, and therefore /(«*) is infinite, then f(z) is also infinite, and so « is a singularity off{z) : the point «=■ - 1 is therefore a singularity of /(«). Similarly since we see that if « is such that z*— 1, then zis & singularity oif{z) ; and in general, any root of any of the liquations «8=1, 5*-l, «8-l, «i«-l, ..., is a singularity of f(z). But these points all lie on the circle |«| = 1 ; and in any arc of this circle, however small, there are an infinite number of them. The attempt to carry out the process of continuation will therefore be frustrated by the existence of this unbroken front of singularities, beyond which it is impossible to pass. In such a case the function f(z) does not exist at all for points z situated outside the circle \z\»l ; the circle is said to be a limiting circle for the function. 42. The identity of a function. The two series 1 + ^-1- ^« + ^+... and - 1 + (2: - 2) - (-2 - 2)« + (-? - 2)» - (^ - 2)* + . . . are not simultaneously convergent for any value of z, and are distinct expansions. Nevertheless, we generally say that they represent the same functiony on the strength of the fact that they can both be represented by the same rational expression . This raises the question of the identity of a function. Under what circumstances shall we say that two different expansions represent the sa/me function ? We shall define a function, by means of the last article, as consisting of one power-series together with all the other power-series which can be derived from it by the process of continuation. Two diflferent analytical expressions will therefore be regarded as defining the same function if they represent power-series which can be derived from each other by continuation. It is important to observe that a single analyticcU eicpression can represent different functions in different parts of the plane. This can be seen from the following example. 60 THE PROCESSES OF ANALYSIS. [CHAP. IH. Consider the series The sum of the first n terms of this series is 1 / 1\ 1 The series therefore converges for all finite values of z. But since when n is infinitely great, z^ is infinitely small or infinitely great according as | -? | is less or greater than unity, we see that the sum to infinity of the series is z "when |-2r|<l, and - when 12:|>1. This series therefore represents one z fwnction at points in tlie interior of the circle | ^r | = 1, and an entirely different ftmction at points outside the same circle. Example. Shew that the series « + g2 + ^+ ,^2+ ... , 1 / 2z 2 1 / 2zy .241 / 2g V \ 2\l-^ 3"3- Vl-«V 3;6-6' Vl-zV -J represent the same function in the common part of their domain of convergence. 43. Laurent's Theorem, A very important extension of Taylor s Theorem was published in 1848 by Laurent ; it relates to the expansion of functions under circumstances in which Taylor s Theorem cannot be applied. Let C and C" be two concentric circles of centre a, of which C is the inner; and let f{z) be a function which is regular at all points in the ring-shaped space between C and G\ Let a + A be any point in this ring-shaped space. Then we have (§ 37, Corollary) y(. + A) l.f -Af) ,, l_.r _/(?L d,, •^ ^ ^mjcz — a — h 2'mJcZ^a — h where the integrals are supposed taken in the positive or counter-clockwise direction round the circles. This can be written + . 43] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 61 We find, as in the proof of Taylor's Theorem, that [ f{z)dz,h^^' f f(z)dz{z-a)^ tend to zero as n increases indefinitely ; and thus we have b b where a^^^f^I^, and K ^ ^. f J^ - aT'^ (z) dz. This result is Laurent* 8 Theorem; changing the notation, it can be expressed in the following form : If z be any point in the ring-shaped space within which f{z) is regular, and which is bounded by the two concentric circles C and C" of centre a, then f (z) can be expanded at the point z in the form h h An important case of Laurent's Theorem arises when there is only one singularity within the inner circle C", namely at the centre a. In this case the circle C can be taken to be infinitely small, and so Laurent's expansion is valid for all points in the interior of the circle C, except the centre a. Example 1. Prove that 1 f^^ where •^n(^)=5- I coB{n6^xsmB)d3. For Laurent's Theorem gives z z where 0^=5^. f Z^*"^^ :^, and bn=J-. [ J^"'^ ^'^dz, and where C and C" are any circles with the origin as centre. Taking C to be the circle of radius unity, and writing z^e , we have 2trtyo = — I cos (nd - ^ sin d) cW, 62 THE PROCESSES OF ANALYSIS. [CHAP. HI. since the parts of / sin (n^-a;sin 6)dB which arise from B and ^ir-6 will destroy each other. Thus Now ft»= (-!)*«»> since the function expanded is unentered if -- be written for z, z Thus 5«=(-i)"y,(x), which completes the proof. Example 2. Shew that, in the annulus defined by \a\<iZi<\bU the expression -j , . yr — x[ can be expanded in the form where 5.- S^ i^.n-:mi^. '[l) • For by Laurent's Theorem if C denote the circle |«|«»r, where |a|<r<|6|, then the coefficient of z* in the required expansion is Putting z=re^f this becomes or 2.r j, * '^ '^l 2*.*! -p-? 201 H~- The only terms which give integrab different from zero are those arising from k^l+n. So the coefficient of z^ is 1 p*^^ 1.3. ..(2^-1) 1.3... (2 ^ +2n-l) <^ Sirjo I 2'.^! 2'+-.(";+n)! &'+»' Similarly it can be shewn that the coefficient of -- is S^a^, Example 3. Shew that e'**'*"«-ao+«i«+«8«*+— +- + J+...> where *'»*^2ir / ^"■*"*'^*^*<^{(^-*^)8"^^-w^}^> and ^»=o^- I ^'''^^'''*^ooa{{v-' u) sine -n6}de. ^ir J Q 44] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 63 44. The nature of the singularities of a one-valued function. Consider now a function f{z) which is regular at all points of a certain region in the -e-plane, except a point z = a;so that the point a is a singularity of the function /{z)* Surround the point a by a small circle 7, with a as centre. Then in the ring-shaped space between 7 and some larger concentric circle (7, the function / (z) can by Laurent's Theorem be expanded in the form ^0 + -4 1 (^r — a) + ^a (-2^ - a)* + -4 , (^ — a)* + . . . z --a {z — ay (z — ay The terms in the last line are called the Principal Part of the expansion of the function at the singularity a ; if they were non-existent, the function would clearly be regular at the point ; so they may be regarded as consti- tuting the analytical expression of the singularity. Now these terms of the Principal Part may be unlimited in number, i.e. the series Si B^ Bi z — a {Z'-ay {z — ay may be an infinite series ; in this case the point a is said to be an essential singularity^ of the function /(z). Or on the other hand, they may be limited in number, i.e. the series just written down may be a terminating series ; so that the expansion can be written in the form Bn Bn-i _^ ^_Bi,^A, + Ai{z''a)'\'A2{z^ay+.... {z-aY {z-af^^ '" z-a In this case the function is said to have a pole of order n at thelpoint a. When n is unity, so that the expansion is of the form — ^H-ilo-hili(-?-a)-hil,(-^-a)"+..., z — a the singularity is said to be a simple pole. Example 1. Find the singularities of the function c z Near 2=0, the function can be expanded in the form e CM afi ^ a a^ a* * The name essential singularity is also applied to any singularity of a one-valued ftmotion which is not a pole, i.e. to singularities for which no Laurent expansion at all can be found. 64 THE PROCESSES OF ANALYSIS. [CHAP. III. c e~ or — ^ e « ( - + c, ) +positive powers of z. There is therefore a simple pole at 2=0. Similarly there is a simple pole at each of the points ^trnia {n= +1, +2, +3, ...). Near 2=0, the function can be expanded in the form c gt-a B-a 14.-1-+ - + or ^ e (>^"-?--)-' ' which gives an expansion involving all positive and negative powers of {z - a). So there is an essential singularity at 2= a. There is also an essential singularity at 2=00 , as will be seen after the explanations of the next article. Example 2. Shew that the function defined by the series , >^-'{(i+y-i ' has a simple pole at each of the points / 1\ ^"^ z^il + -\e * (it«0, 1,2, ...7i-l; ?i=l, 2, ...00). (Cambridge Mathematical Tripos, Part II., 1899.) 45. The point at infinity. The behaviour of a function f(z) for infinite values of the variable £ can be brought into consideration in the same way as its behaviour for finite values of z. For write ^ = -> , so that the infinite values of z are represented by the point / = in the /-plane. Let f(z)^<f> (z). Then the function ^ (/) may have a zero of order m at the point 2:' = ; in this case the Taylor expansion of ^ (/) will be of the form and so the expansion oi f{z) valid near z^(x> will be of the fonn T" • • • • In this case, / (z) is said to have a zero of order m at z==oo , 45] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 65 Again, the function ^ (/) may have a pole of order m at the point / = ; in this case, and so for large values of z,f{z) can be expanded in the form In this case, ^ = oo is said to be a pole of order m for the function /{jb). Similarly f(z) is said to have an essential singularity at -^ = oo , if ^ {£) has an essential singularity at the point z = 0. Thus the fiinction eF has an 1 essential singularity at 5 » oo , since the function ^ or has an essential singularity at / = 0. Example, Discuss the function represented by the series The function represented by this series has singularities at z^--^ and f~ — -y (n«l, 2, 3, ...), since at eacb of these points the denominator of one of the terms in the series is zero. These singularities are on the imaginary axis, and are infinitely numerous near the origin ^=0 : so no Taylor or Laurent expansion can be formed for the function valid ia the region immediately surrounding the origin. For values of z other than these singularities, the series converges absolutely, since the ratio of the (n+l)th term to the nth is ultimately . > ^ , which is veiy small when n is larga The function is an even function of z (i.e. is unchanged if the sign of ^ be <:hanged}, is zero for all infinite values of 2, and m regular at all points outside a circle C of radius unity and centre at the origin. So for points outside this circle it can be expanded in the form -where, by Laurent's Theorem, ** 2ni J a n=ow^ «"■** + «* 0-a«^tt-l ^»-3^-2«/ a-2n ^-4n ^-e» \ ^^"^ n\(a'^+z^)" ^1 V""i5~ "*""?"" 15-+-J' 1 (-!)*-! a"** and the coefficient of - on the right-hand side of this equation is ^ — -. . Z 7Cr • W. A. 5 66 THE PROCESSES OF ANALYSIS. [CHAP. lU. Therefore, since only terms in - can fiimish non-zero integrals, we have *ifc=o— • ^ I — i ^ 2mn^oJc n\ z j_ =(-l)*-ie«". Therefore for large values of z (and indeed for all points z outside the circle of radius unity) the function can be expanded in the form JL J. J_ The function has a zero of the second order at z== ao , since the expansion begins with a term in -5 . z^ 46. Many-valued functions. In all our previous work we have supposed the function f(z) to have one definite value corresponding to each value of z. But functions exist which have more than one value corresponding to each value of z. Thus the function ^ has two values (viz. + VJ and — V^) corresponding to each value of z, and the function tan~* z has an infinite number of values, expressed by the formula tan~^ z ± hrr, where k is any integer. We may however for many purposes consider + Vz and — VJ as if they were two distinct functions, and apply to either of them separately the theorems which have been investigated in this chapter. When we in this way select some one determination of a many- valued function for considera- tion, it is called a branch of the mimy-valued function. Thus the values log z, log z + 27rt, log z + 4iTn, . . . , would be said to belong to different branches of the function log z. There will be certain points for which the values of the function given by diflferent branches coincide: these points are called branch-points of the function, and must be included among its singularities. Thus the function jgr* has a branch-point at jgr = 0, since either branch there gives the same value, zero, for the function. It must not however be supposed that the branches of a many-valued function really are distinct functions. The following example shews how the different branches of a many-valued function change into each other. Let f{z) = A 46] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 67 Write 2:=r(co8^ + i8ind), where 0<^<27r. Then the two values of f{z) are e . . e\ , ,-/ . . . 0' + Vr [cos ^ + i sin 5] and - Vr [cos ^ + i sin 5 j . Let us take the former of these values, and consider its changes as the point z describes a circle round the origin {z = 0). As the point travels, r is unchanged, but constantly increases, and when the point reaches again the starting-point after completing the circuit, has increased by 27r. The function has therefore become r { + 27r . . ^ + 27r\ -h Vr (cos — 2 — + ^ sin — — j , ^r — Vr (cos^ + isin ^j. In other words, the branch of the function with which we started has passed over into the other branch. In following the succession of values of f{z) along a given path, the final value is deduced without ambiguity from the initial value; and all con- ceivable paths lead to one of two final values, viz. 'Jz and — Vz, But it appears from the above that it is not possible to keep these branches per- manently apart as distinct functions, because paths lead from one value to the other. The idea of the different branches of a function helps us to understand many of the ''paradoxes'' of mathematics, such as the following. Ck)nsider the function du for which t- =^ (1 +log «). When z is negative and real, -^ is not real. Now if ^ is a negative quantity of the form \^ (where p and q are positive or negative integers), u is real. If therefore we draw the real curve we have for negative values of « a series of conjugate points, arranged at infinitely small intervals of z : and thus we may think of proceeding to form the tangent as the limit of du the chord, just as if the curve were continuous ; and thus -r- , when derived from the inclination of the tangent to the axis of a?, would appear to be real The question thus dtt arises, Why does the ordiniiry process of differentiation give a non-real value for -r- ? The 5—2 68 THE PROCESSES ^OF ANALYSIS. [CHAP. IH. explanation is, that these conjugate points do not all arise from the same branch of the function u=^. We have in fact and log z has an arbitrary additive p£urt 2lrirt, where k is any integer. To each value of k corresponds one branch of the function u. Now in order to get a real value of u when z is negative, we have to choose a suitable value for k : and this value ofk varie$ atvfego from one conjugate point to an adjacent one. So the conjugate points do not represent values of u arising from the same branch of the function ubs^*, and consequently we cannot expect diL the value of -r- to be given by the tangent of the inclination to the axis of x of the tangent-line to the series of conjugate points. Example 1. If log z be defined by the equation log i?a Limit n{f^—\\ »=• shew that log 2 is a many- valued function, which increases by 2n-t when z describes a closed path round the origin. For put «=r (cos 6-\-imi 6). Then one of the vcJues of log 2, on this definition, is Limit n if* (cos — Hisin- |-lL »=■« I \ n M/ , J ' 1 where r* is the positive nth root of r. This can be written 1 Limit n {r* - 1} + i6, naoo « When z describes a closed path round the origin, the first term in this expression remains unaltered, while the second increases by 2frt ; hence the result. Example 2. Find the points at which the following functions are not regular. {a) «*. Answer, «— 00. (6) cosecs. Answer, z=0, ±ir, ±2ir, ±3ir, .... z-\ ^^^ «»-5«+6- Answer, z^2,Z, 1 {d) e\ Answer, «=0. (6) {(^-1)4*. Answer, «=0, I, 00. Example 3. Prove that if the different values of a», corresponding to a given value of z are represented on an Argand diagram, the representative points will be the vertices of an equiangular polygon inscribed in an equiangular spiral, the angle of the spiral being independent of a. (Cambridge Mathematical Tripos, Part I., 1899.) 47, 48] THE FUNDAMENTAL PEOPERTIES OF ANALYTIC FUNCTIONS. 69 47. LtouvUle's Theorem. We know by § 38 that if /(^) be any function of z which is regular at all points of the ir-plane within a circle G, of centre a and radius r, then ^ ^^^ 27^tjc(^-a)«+^• Now let M be the greatest value of \f(z)\ at points on the circle 0. Then this equation gives (§ 32) nlM From this inequality an important consequence can be deduced. Suppose that /(z) is, if possible, a regular function of z over the whole z-plane, including infinity, ie. that it has no singularities at all. Then in the above equation M is finite when r is infinite, whatever n is ; and therefore /<*^ (a) is zero for all values of n and a, i.e. /(a) is a constant independent of a. We thus arrive at Liouville*8 theorem^ that the only fwnction which is regular everywhere is a constant. As will be seen in the next article, and again frequently in the latter half of this yolnme, Liouville's theorem furnishes short and convenient proofis for some of the most important results in Analysis. 48. Functions with no essential singularities. We shall now shew that ^ ofnly one-valued fu/nctions which have no singularities in either the finite or infinite part of the plane, except poles, are roMonal functions. For let / (z) be such a function ; let its singularities in the finite part of the plane be at the points Ci, Cj, ... c*: and let the principal part (§44) of its expansion at the pole Cy be Z-'Cr (z^CrY (^-Cr)"^ Let the principal part of its expansion at the pole £: = oo be if 5 = 00 is not a pole, but a regular point for the function, then the coefficients in this expansion will be zero. Now the function ^^^ r%\z-Cr^iz-Cry^-^{z-Crrr] 70 THE PROCESSES OF ANALYSIS. [CHAP. III. has clearly no singularities at the points Oi* ^> ••• Cjb, x ; it has therefore no singularities at all, and so by Liouville's theorem is a constant ; that is, f{z) = constant 4- Ojir + (V* + ... + (ti^ f{z) is therefore a ratioual function, and the theorem is established. Miscellaneous Examples. 1. Obtain the expanBion 2. Obtain the expansion /(.)-/W+"i^°[/-W+/'(.)+!{/'(.+^)+/'(»+^)+- + .... (Corey.) 3. Obtain the expansion + ... (Corey.) 4. In order that values U-\- Vt, which are given as continuous functions of the arc of a circle, should be the boimdary values of an analytic function, shew that it is necessary and sufficient : (a) That — '^ — ^^^ — — — -^ at the place ^=0 should be uniformly integrable for all values of a ; {h) That the values of 7 shall be given by V{a)^^ i' {U{a'-^)'U{a^^)}Qot^d^. (Tauber.) MISC. EXS.] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 71 5. Shew that for the series «i»0 2»+«-*' the region of convergence consists of two distinct areas, namely outside and inside a circle of radios unity, and that in each of these the series represents one function and represents it completely. (Weierstrass.) 6. Shew that (Jaoobi & Scheibner.) 7. Shew that +*"<"*+^^-;-^'"+**>(i-»)-'/'V(i-<)«'-t dt. (Jacobi & Scheibner.) '8. Shew that (i-^-.;^.,.-«-.*.=i;{,.=±|.*....<:-:|tg*-;)^-.} ^c--'-' ,i:t>y.y/s.':'j :'--<'-'-'-''"- (Jacobi & Scheibner.) 9. I( in the expansion of (a+Oi^+tv')*^ by the multinomial theorem, the remainder after n terms be denoted by i^ so that (a+ai«+a^«)«»=ilo+i4i2+ii22*+...+-4n-i«*"^+^ shew that i2»(a+ai*+<V«)-j^ (^^^^^^pri <«• (Jacobi & Scheibner.) 10. If (ao+ai«+a^)-*-* j{ao+a^t+a^t*)'^dt be expanded in ascending powers of z in the form shew that the remainder after (n— 1) terms is (Jacobi & Scheibner.) 11. Shew that the series 2 {l + X,(.)e'}-^jr-, nao where X,»(«)= -l+^-gj + ^j -...±^, 72 THE PROCESSES OF ANALYSIS. [CHAP. III. and where 4> (^) is a regular function of z near «aO, is convergent in the neighbourhood of the point «=0; and shew that if the sum of the series be denoted bj /{$), then /(f) satisfies the differential equation /' {e) -/(«) - (z). (Pincherie.) 12. Shew that the arithmetic mean of the squares of the moduli of all the values of the series 2 Oj^^ on ^ circle |f|»r, situated within its circle of convergence, is equal to the sum of the squares of the moduli of the separate terms. (Qutzmer.) 13. Shew that the series 00 converges when |f | < 1 ; and that the function which it represents can also be represented when \z\ < 1 by the integral /a\4 /■• $'» dx \w/ Jo' ^-z X*' and that it has no singularities except at the point f ~ 1. (Lerch.) 14. Shew that the series 2 2 ( z «-i 1 - (* + ^*) + - 2 l^j _ 2^_ 2^,^.^ (2v+ 2v'«)« ■*■ (1 - 2^- 2^z-H') (2p+2p'z'hyi ' in which the summation extends over all integral values of p, i/, except the combination (iraaO, V »0), convergcs absolutely for all values of z except purely imaginary values ; and that its sum is '+ 1 or - 1, according as^the real part of z is positive or negative. (Weierstrass.) 15. Shew that sin •Ite ( « + -H can be expanded in a series of the type Z Zr in which the coefficient of either t^ or z'^ia 5-1 sin (2tt00s ^) cos nBdB, 16. If hew that /(f) is finite and continuous for all real values of f, but cannot be expanded as Maclaurin's series in ascending powers of z ; and explain this apparent anomaly. CHAPTER IV. Thb Uniform Convergence op Infinite Series. 49. Uniform Convergence. We have seen* that the sum of a convergent series of analytic functions of a variable z can have discontinuities as z variea It was found by Stokes "f and Seidell in 1848 that this can never happen except in association with another phenomenon, that of non-uniform convergence, which will now be investigated. Consider the series « ?! . ^ + ^(2^-1) ( 1 + 2^) ( 1 + 2^ + ^2) "^ ( 1 + 2z + ^) ( 1 + 3^ + ^) "^ • • • . 5 + g^U-l) We shall first shew that this series is convergent for all values of z except certain isolated points. For, except for the roots of 1 + w-r + ^ = 0, the nth term can be put in the form 1 1 l+n^ + -8^ l+(n + l)^H- sf^-^^ ' so the sum of the first n terms is o 1 1 **- 1 + 2z i + (n + 1)« + -2*+* ' which, as n becomes infinitely great, tends to the value ^ ^ for all points except 5 = 0: and for -r = 0, we have /S = 0. Thus (except at the roots of the equations I + nr + ^ = 0) the series converges ; and it represents a regular function, except at ^ = 0, where it has a discontinuity. * In § 42. t Collected Paper$, Vol. i. p. 286. t Miineh. Ahh. 74 THE PROCESSES OF ANALYSIS. [CHAP. IV. What lies at the root of the discontinuity ? The remainder after n terms is For ordinary values of z, say ^ = 1, this remainder decreases rapidly as n increases. Thus if w = 10, j? = 3, the remainder = o j. . on > * negligible quantity. But now let z approach near to its discontinuity 0: say ^~ 1000000* "r^®^ ^i^'^ ^^^ value of z, the remainder after 1000 terms is nearly 1, and the remainder after 1000000 terms is still nearly s- This shews that, as z approaches the discontinuity at ^=»0, the terms which contribute sensibly to the sum tend to recede to the infinitely distant part o/ the series, so the first 1000 terms do not furnish a good approximation at all. We can express this analytically as follows : — The number of terms n, which we have to take in order to make |i2n| less than a given small positive quantity e, tends to oo as we approach the point of discontinuity. This circumstance is the basis of the following definition : — Let Ui{z)-\- u^{z)'\-Ut{z)-\-u^{z)'\- ,.. be a series of functions of z, which is convergent at all points z within a given area in the j?-plane. Let Rn be the remainder after n terms. Then since the series converges, if we take a small finite quantity e we can find at any point on the area a number r (varying from point to point) such that \Rn\ < € so long as n > r. If the numbers r corresponding to the aggregate of points in the vicinity of a given point z are all less than some definite finite number, the series is said to be uniformly convergent at the point z ; but if near any point z the number r tends to infinity, so that no definite upper limit can be assigned to it, the convergence of the series is said to be non-uniform* in the neighbourhood of the point z. Example 1. Shew that the series ■*■ 1 +a* "^ (1 +««)«'^ •'• ■*"(TT?)* "*"'"• * which converges absolutely for all real values of is, is discontinuous at z—O and is non- uniformly oonvei*gent in the neighbourhood of ^— 0. The svun of the first n terms is easily seen to be 1 +«*— ,-^ — sxr;:^ . So when z is not zero the siun is 1 +^, and when z is zero the sum is zero. * An interesting geometrical treatment of uniform convergence is given by Osgood in VoL lu. of the BvU. of the Amer, Math, Soe, p. 59 (1896). 49] THE UNIFORM CONVEROENCE OF INFINITE SERIES. 75 The remainder after n terms is 7rTT2\»=i* "^^^ ^^^^ ^ made smaller than any log- aasigned smaU finite positive quantity c by choosing n so that n-l> j j m . But as 9 tends to aero, . 7|^r^\ tends to infinity, so n must tend to infinity, i.e. we have to include an infinite number of terms in order to get the remainder less than c . This series is therefore non-uniformly convergent in the neighbourhood of e=0. Example 2. Shew that at «=0 the sum of the series z z z l(z+l) ' (a+l)(25+l)"-^{(n-l)«+l}{fw+l} is discontinuous and the series is non-uniformly conveigent. The sum of the first n terms is easily seen to be 1 ^ : so when z is zero the sum is 0. The remainder after n terms of the series is — — : ; so when z is nearly zero, say fs one-hundred-millionth, the remainder after a million terms is --^^ or 1 - =^ , so 100^ the first million terms of the series do not contribute one per cent of the sum. And in general if i be small, it is necessary to take n large compared with the large quantity - in order to make the remainder after n terms smalL There is therefore non-uniform z convergence in the neighbourhood of 2=0. Example 3. Discuss the series »=i{l+»M{l-Kn+iy«««}- The nth term can be written ; 5-^ - _— ^ Tr^— , , so the sum to infinity is r— — 5 , l-»-nV l-|-(n-|-l)'«* l-l-«* and the remainder after n terms is z— ^ tth-, . 1-H(n-|-1)*2* However great n may be, i^we take z equal to -tt> ^^^ remainder will have a finite value, namely \ ; the series is therefore non-uniformly convergent at 2^0. NoTB. In this example the sum of the series is not discontinuous at ««>0. Cayley* regards non-uniform convei*gence as consisting essentially in the occurrence of a discontinuity in the sum of a series. The condition for a discontinuity in a series at the point z^av& that the series T^\ a-z shall have an indefinitely large sum when (a— z) is indefinitely small. * ** Note on Uniform Oonvergeaoe," Proc, Hoy, 80c. Edinb. xix. (1891-2), pp. 208-8. 76 THE PROCESSES OF ANALYSia [CHAP. lY. Thus in the series (l-^)+«(l-;?)+z«(l-«)+..., which is non-uniformly convergent and discontinuous at ^a*!, we have a — z = -«*, when a=l, 80 the sum of the series 2 ^^-^ is ; — , which is infinite for «= 1. 60. Connexion of discontinuity with non-uniform convergence. We shall now shew that ths sum of a series of continuous functions of z, if it is a uniformly convergent series for values of z within certain limits, cannot he discontinuous for values of z within those limits. For let the series be /(2r) = 1^1 (^) + 1^3 (<2:)+...+ 14^ (2r)+... = flfn(^) + lJnW> where B^ is the remaiDder after n terms. Since the series is uniformly convergent, we can to any small positive number e find a corresponding integer n independent of z, such that I iZn (-3^) I < « for all values of z within the area^. Now n and € being thus fixed, we can, on account of the continuity of Sn (z), find a positive number 17 such that, when |5 — / 1 < 17, the inequality is satisfied. We have then <\Sn(z)''Sn(z')\ + \Rn(z)\'^\Itn{z')\ which establishes the fiesult. Exam^ 1. Shew that at «=0 the series 1 1_ where Wj («)==«, «»(«)=«*•"*-«*•"•', and real values of z are concerned, is discontinuous and non-uniformly convergent 1 The sum of the first n terms is i^~^ ; as n tends to infinity, this quantity tends to 1, 0, or - 1, according as 2 is positive, zero, or negativa The series is therefore absolutely convergent for all values of z, and has a discontinuity at f =0. 1 The remainder after n terms, when z is small and positive, is 1 -i^'^ ; however great n may be, by taking «=c~(*»~i) we can cause this remainder to take the valuie 1 - - , which is different from zero. The series is therefore non-uniformly convergent at «>bO. 50, 51] THE UNIFORM CONVERGENCE OF INFINITE SERIES. 77 ExamjiU 2. Shew that at 2=0 the aeries n-i{H-(l+^)*-i}{l +(!+«)*} is discontinuous and non-uniformly conveigent. The nth term can be written l-(l+g)* _ l-.(l+g)*-i l+("l+2)"* l+(l+z)»-i' the sum of the first n terms is n . /i , w • Thus considering real values of t greater than - 1, it is seen that the stun to infinity is 1, 0, or - 1, according as 2 is negative and greater than -2, zero, or positive. There is thus a discontinuity at 2«>>0. This discon- tinuity is explained by the fact that the series is non-uniformly convergent at «=0 ; for the remainder after n terms in the series when t is positive is -2 and however great n may be, by taking e=- this can be made to take the value — 2 , which is difilerent from zero. The series is therefore non-uniformly convergent so 1+e at «=0. 61. Distinction between absolute and uniform convergence. The uniform convergence of a series does not necessitate its absolute convergence, nor conversely. Thus the series (§ 49, Ex. 1) S .^ ^.^ con- verges absolutely, but (at ^ = 0) not uniformly : while if we take the series « (, l)n-l its series of moduli is which is divergent, so the series is only semi-convergent ; but for all real values of 2, the terms of the series are alternately positive and negative and numerically decreasing, so the sum of the series lies between the sum of its first n terms and of its first (n + 1) terms, and so the remainder after n terms is less than the nth term. Thus we only need take a finite number of terms in order to ensure that for all real values of z the remainder is less than any assigned quantity, i.e. the series is uniformly convergent. Absolutely convergent series behave like series with a finite number of terms in that we can multiply them together and transpose their terms. Uniformly convergent series behave like series with a finite number of terms in that they are continuous and (as we shall see) can be integrated term by term. 78 THE PROCESSES OF ANALYSIS. [CHAP. IV. 62. Condition /or uniform convergence, A sufficient though not necessary condition for the uniform convergence of a series may be enunciated as follows : — If for all values of z within a certain region the moduli of the terms of a series 8 = v^{z) -k- v^{z) '\- u^ W + ••' are respectively less than the corresponding terms in a convergent series of positive constants then the series 8 is uniformly convergent in this region. This follows from the fact that, the series T being convergent, it is always possible to choose n so that the remainder after the first n terms of T, and therefore of 8, is less than an assigned positive quantity € ; and since the value of n thus found is independent of z, the series 8 is uniformly convergent. GoroUary. The theorem is still true if the moduli of the terms of <S, instead of being less than the terms of T, are to them in a variable but finite ratio. Example. The series COSZ+gjCOS*;? + ^COS'« +... is uniformly convergent for all real values of z, because the moduli of its terms are not greater than the corresponding terms of the convergent series 1+^+1+ whose terms are positive constants. 63. Inteffration of infinite series. We shall now shew that if 8(z)=^iii(z) + ti^{z)-\' ... is a uniformly con- vergent series of continuous functions of z, for values of z contained within some domain, then the series jtifi(z)dz'\-jv^(z)dz + ... , where all the integrals are taken along with some path C in the domain, is convergent, and has for sum l8(z) dz. For let n be some definite finite number, and write 8 (z)=^ U,(Z) + U^(z)-¥ ... '\' Un(z) + Rn{z), so j 8(z)dz== jtii(z)dz+...'^ Iun(z)dz + lRn(z)dz. 52, 53] THE UNIFORM CONVEBGENCE OF INFINITE SERIES. 79 Now since the series is uniformly convergent, to every positive number € there corresponds a number r independent of z, such that when n^ r we have R^{z)\< €, for all values of z in the area considered. Therefore if 2 be the length of the path of integration, we have (§ 32) /' <d. Rn{z)dz Therefore the modulus of the difference between I S(z)dz and the sum of the n first terms of the series X I Un (z) dz is less than any positive number provided n is large enough. This proves both that the series 2 jt^(z)dz is convergent, and that its sum is J8(z)dz, Example 1. As an example of the necessity of this theorem, consider the series • 2z{n(n + l)sin*«2- l}co8«' 2 ««i {l+n''sin2«8}{l + (n+l)«8in««2}* The nth term is 2zn cos 2* 2z(n+l) cos z* l+n«sin2 2«"' ! + (« + !)* sin^z*' and the sum of n terms is therefore 2gcosg' 2g(n+l)cosg* The series is therefore absolutely convergent for all real values of ;; : but the remainder after n terms is 2z{n+l)co8z'^ H.(n+l)2sin2?' and if n be any number however infinitely great, by taking z= — -i this has the finite value 2. The series is therefore non-imiformly convergent at ;?«0. Q* COS Z^ Now the sum to infinity of the series is i— .—5-,, and so the integral from to « of the sum of the series is tan~^ (sin ^). On the other hand, the sum of the integrals from to i? of the first n terms of the series is tan~* (sin a^) - tan** (n-fl sin «*), and foms= 00 this tends to tan~*(sin«')— ^. Therefore the integral of the sum of the series differs frx>m the sum of the integrals of the terms by •= . Example 2. Discuss the series • 2g^{l-n(e-l)+e*'^^^} ,2in(n+l)(l+^«^)(l+e»+»««) for real values of z. 80 THE PROCESSES OF ANALYSIS. [CHAP. IV. The nth term of the series may be written n(l+e^«) (n+l)(l +«*+»««)• The sum of the first n terms is 1+^" (71+ 1)(1 +€•*!««)• The series therefore converges to the value ^ ; and since the terms are real and ultimately of the same sign, the convergence is absolute. The integral from to « of the sum of the series is log (1 +«?«). The sum of the first n terms of the series formed by integrating the terms of the series is log(l+e0«)-^log(l+e»*ii^), which fornsoo tends to l0g(l+6««)-l This discrepancy is accounted for by the non-uniform convergence of the series at f «0 ; for the remainder after n terms in the original series is or (n+l)(l+«-^) «_+?,-t+(n+i); i and however great n may be, on taking «=> -— -r this takes the value unity ; so the series is non-uniformly convergent at f ^0. Example 3. Discuss the series W1+W8+W3+..., where Uj ■= ze-'^f tt» = nw-"** - (n - 1) -w^*-*)**, for real values of z. The sum of the first n terms is nzer*^, so the sum to infinity is for all real values of z. Since the terms Un are real and ultimately all of the same sign, the conveigenoe is absolute. In the series / riidz+l t^dz+ I i(^dz-\',„f Jo Jo Jo the sum of n terms is ^ (1 - 0"**^), and this tends to the limit ^ as n tends to infinity ; this is not equal to the integral from to i? of the sum of the series 2 u^. The explanation of this discrepancy is to be found in the non-uniformity of the convergence neariS^O, for the remainder after n terms in the series Wj+Wj-I-... is -Twe-*^ ; and however great n may be, by taking 2= - we can cause this to tend to the limit - 1, which is different from zero: the series is therefore non-imiformly convergent near z=0. 54, 55] THE UNIFORM CONVERGENCE OF INFINITE SERIES. 81 64. Differentiatimi of infinite series. The converse of the last theorem may be thus stated : If 8 (z) ^ Ui(z) -h u^{z) + , .. is a convergent series of analytic functions of z, which are regular when the variation of z is restricted to be within a certain domainyandiftheseries'S.{z)^-r'V^{z)-\--j-ii>2(z) + ... is wniforrrdy convergent within this domain, then this latter series represents -j- 8 (z). For by the preceding result, if a and z are two points within the domain, we have r X(t)dt = r ui if) dt + \\i (t)dt+,.. J a J a J a Since if^(z) + u^(z)+ ,,. and i^i(a) + ti,(a) + ... are each of them convergent series, we can write this J c ^{t)dt = {i^i (z) + iLi{z) + .,.]-'{ui (a) + Wa(a) + ...} = 8(z)^8(a), and hence \ We may note that a derived series may be non-imiformly convergent even when the original series is uniformly convergent : for instance the series sin«— J8in2«+Jsin3«+.., is non-uniformly convergent at z=^Tr; although the series from which it can be derived, namely - cos a+5jCOS 22- ^ cos 3«+ ..., is uniformly convergent for all real values of z. 66. Uniform convergence of Power-8eries, We shall now shew that a power-series is wniformly convergent at all points within its circle of convergence. For let jR be a region, forming part of the area of the circle, and let r be a quantity greater than the modulus of every point of i2, but less than the radius of convergence. Then if z be any point of R, the moduli of the terms of the series ao + aiZ + a^-^- ... w. A. 6 82 THE PROCESSES OF ANALYSIS. [CHAP. IV. are less than the moduli of the corresponding terms of the convergent series ao + ctir + a,i'*+ ... . But the latter series does not involve z, and so (§ 52) the power-series is uniformly convergent within the region R\ as R is arbitrary, the series there- fore converges uniformly at all points within the circle of convergence. It must be observed that nothing is proved regarding points on the circumference ; we do not even know that the series is convergent there at all. Corollary, A power-series is continuous within its circle of convergence : and the series obtained by differentiating and integrating it term by term are equal to the derivate and integral of the function respectively. Example, As an example of this, consider the series which is convergent at all points within a circle of radius 1. We can integrate it term by term, so long as the path of integration lies in this circle ; the result is /; Now / T-T-h clearly represents that value of tan~i» which lies between - ^ *^d +^ , So the series represents this value of tan" ^« and no other. Miscellaneous Examples. 1. Shew that the series represents . .^ when \z\<l and represents >, when |«|>1. Is this fetct connected with the theory of uniform convergence ? 2. Shew that the series 2sin- +4sin;r +...+2*sin ;;— + ... converges absolutely for all values of z, but does not convei^ uniformly near «bO. 00 3. If aserie8^(«)«r j (c^- c,,+j) sin (2i» + !)««• (in which Cq is zero) converges uni- IT formly in an interval, shew that g{z) -. is the derivate of the series /(«)= 2 — sin2i«ir. (Lerch.) CHAPTER V. The Theory of Residues : Application to the > Evaluation of Keal Definite Integrals. 66. Residties. If a point ^ = a is a pole of order m for a function /(z), we know by Laurent's theorem that the expansion of the function near 2r = a is of the form where ^ (s) is regular in the vicinity of ^ = a. The coeflScient a»i in this expansion is called the residue of the function f{z) relative to the pole a. Consider now the value of the integral \f{z) dz, where the integration is taken round a circle 7, whose centre is the point a and whose radius is a small quantity p. We have I f(z)dz=l, a^r f 7 ;z + / <l>{z) dz. Now I <l> {z) rf^ = 0, since <f>(z)iB a regular function in the interior of the circle 7 : and (putting z — a^ pe^) we have , = p-»^-i r^ , when r + 1 = 0, when r^\. But when r=! I we have f J^ = ride = 2^. jyZ — a Jo Hence finally / f(z) dz = 2'rna^i. Jy Now let C be any contour, containing in the region interior to it a number 6—2 84 THE PROCESSES OF ANALYSIS. [CHAP. V. of poles a, 6, c, ... of a function /(^), with residues a_i, 6_i, c_i, ... respec- tively : and suppose that the function f{z) is regular at all points in the interior of (7, except these poles. Surround the points a, 6, c, ... by small circles a, /8, 7, ... : then since the function f(z) is regular in the region bounded by (7, a, ^8, 7, ..., its integral taken round the boundary of this region is zero. But this boundary consists of the contour (7, described in the positive sense, and the contours •1 A 7> ••• described in the negative sense. Hence = [ f{z) dz -| f{z) dz-^j f{z) dz..,, or ^ ~ I f{^)^^ ■" 27rwi«i — 27rt6_i ••• • J c Thus we have the theorem of residues, namely f{z)dz^'l7ntR, L C where 2-B denotes the sum of the residues of the function /{z) relative to those of its poles which are situated within the coo tour 0. This is an extension of the theorem of Chapter III. § 36. 67. Evaluation of real definite integrals. A large number of real definite integrals can be evaluated by the use of contour-integrals and the theorem of residues. The following examples will serve to illustrate the various ways in which these aids to the evaluation may be applied. Example 1. To find the values of P%«**co8(n^-sm^)cW and (^ e"^^ sin {n3- am 6) dS, Denoting these integrals respectively by / and •/, we have Write 0^=2, and let (7 be a circle of radius unity round the origin in the f -plana Then «8 6 assumes the sequence of real values from to 2n-, 2; describes the circle C, Hence l-iJ^-. I eFz'^'-^dz ^ J c =s 27r X the residue of -—ry at «=0 _2.r 1 57] THE THEORY OF RESIDUES. 85 Therefore -^■" -r i Example 2. The method used in Example 1 can be very generally applied to trigonometrical integrals taken between the limits and 2v, As another example, consider the int^pral /-r f-, (a>6). Write efi^^z'y and let C be the circle on the z-plane whose centre is at the origin and whose radius is unity. / [ 2cb Then dz » Jc =4ir X simi of residues of j-^ — r. at poles contained within C. Now fea»+2€W+6 2^a^-b*- -1 a V a* — 6* a v a* — 6* Thtt^ore the two poles are at 2= - ^ r^- and f ■■ - ^ + ^ ^ ~ , and the residue at the former (which is the only one within C) is -- — — . Hence /« Example 3. Shew that (a+6cos5)2~(a2-'62)>/2' Example 4. Find the value of i: / * x sin ma? , ax. -00 ^+a* Let (7 be a contour formed by the real ckxis together with a semicircle y, consisting of that half of a circle, whose centre is at the origin and whose radius is very large, which lies above the real axis. Then ^ - j is a function of z which has only one pole in the interior of C, namely at ,-— , dz = 2*rt X residue of -, ^ at its pole at. But writing z^at-^-^y we have ^— ma =-o-v + positive powers of f. 86 THE PROCESSES OF ANALYSIS. [CHAP. V. Therefore ^ ^ = -y^—-ir+ positive powers of («-ai). Thus the residue of -5- — = at at is - «"*•. The«fo« .ir~-/^ -;^,«fc=(/^^ +/J /g,^ Since is infinitesimal compared with pr at points on y, the integral round y is infinitesimal compared with / — Therefore wte"* Equating imaginary parts, we have or 2fr, and is therefore zero. /* iT sin mx , _^ Example 6. To find the value of / ««~"**sin(asin6^)^— ^. Take a contour C composed as in Example 4 of an infinite semicircle y and the real axis. Then / — .«<" ^ . ^ cfe—2»ri x residue of 77-. «** « . • at its poles inside C But r\««« , A has only one pole in the interior of C, namely at the point z^rL Now if 2«an+f, we have 2i^ ?+7i = 2i'^ 2nf+T* = 4^'^ + positive powers off. 1 6r Therefore the residue is t-^' ' But at points on y, e^«*=0, so e<*« =1, and so Jy2i a^+f^ 2tjy^"2* Therefore ^eo^-^^^^./" e^ooBhx^j^^amibx) ^^, or JVoo.ta8m(a8in6*)J;^=^ (««-»'-!). We may note that in the above 1 stands for the limit of 1 where k is infinitely /■* great, and is not equal to the limit of 1 where k and I are different. EXS.] THE THEORY OF RESIDUES. 87 Example 6. Prove by integrating /: ^dz round the contour used in Examples 4 and 5, that Example 7^ 'Find the value of /, * mnmx . Consider a contour C, formed of 1<> a semicircle r whose centre is at the origin and whose radius is very large, 2^ a semicircle y whose centre is also at the origin and whose radius is very small, and 3** the portions of the real axis intercepted between these circles. The semi« circles are to be drawn in the upper half of the «-plane, i.e. the half above the real axis. . , . — 5rs«2frt X the residue of -7-,-:— -v at the singularity Zmeat. But if we write 2= at + (> where C is small, we have Thus the residue at at is - Therefore - 4a» (-!)• ia? (**+a)' jc «(«»+««)« ~U-. "^jr JvJ*(»«+a«)»- Now / a . jNg is infinitely small at points on r, so the integral taken round r vanishes. _co«(2:«+a*)* a* 20* \ a/ In this, I means f + f > where the two c's are the same : but in the final result J -00 J 9 J —00 we can put f >bO, since the final integrand is finite at the origin. Equating the imaginary parts on both sides of this equation, we obtain /* sin mxdx w irg"*^ / 2\ -ooi"(^+a«)«"^" 2a» [^'^aj' J /"• Binmxdx it ire"^^/ . 2\ And so I —7-5 oTo^TT-i 7— r— |w*+-|. Jo^{x^-^ay 2a* 4a» \ a/ Example 8. Find the value of /:. * cos 2ax - cos 26^ , 1^ '^- Take the contour C formed as in Example 7 by an infinite semicircle r, a small semi- 88 THE PROCESSES OF ANALYSIS. [CHAP. V. circle y round the origin, and the parts of the real axis intercepted between them. Within this contour the function — |- has no singularities. In this equation i must be r^arded as an abbreviation for i + i where f is the radius of y. ^8«<* 1 Now at points on r, —^ is ssero compared with - , so the integral round V is zero. z* " z -— cb a one-half of 2tnx the residue of — ,- at the origin am X the residue of = = — 2ira* —^ dz=-27ray — 00 ^ ^ cfe=2fr(6-a). Taking the real part of this we have /* cos 2a:r - cos 26^ j ^ ,, \ ^ flte=2»r(6-a), , . cos 2cu? - cos 2&r . i, .^ i ^ j i _j__- x T* ^ and smoe -^ is nmte when ^=0, we need no longer restnct / to mean Example 9. Find the value of /o'--«^(T-^)^f^ («>0). We have J^ ^-i sin (^ - 6*) ^^ Consider a contour (7, formed as in Examples 7 and 8 by an infinite semicircle r, a small semicircle y round the origin, and the parts of the real axis intercepted between them. Then i j (-«y»-ic««^^=2irtxthe residue of | (-^O^'^^^^T^ft^^ i^ singularity «=ir. EXS.] THE THEORY OF RESIDUES. 89 Putting 2arn+C <^^ n^ecting powers of (y we see that the expansion of begins with a term "4— f— ' 80 the required residue is — j r*-* c-^. Therefore s^"*«"^=5 / (-^t)"-^ z z J c rdz At points on r the integrand is infinitesimal compared with - , and so the integral z round r is zero. At points on y the integrand is approximately ^—^ — z^^ and so if a > the integral round y is zero. Therefore / ^^s- (^-fe-)^-g, = | /^J-^T-.- /* /it: gaooBtegin (a sin 6^) — . ^ Wehave f Voo«to8in(asin6x)~ =i f * «<»^— , Jo 'a; 2i j_ao 4? ' where in the latter int^ral i must be regarded as an abbreviation for i +1 where € is a small quantity. Take a contour C, consisting as in Examples 7, 8, 9, of an infinite semicirole r, a small semicircle y of radius t roimd the origin, and the parts of the real axis intercepted between them. Then 0= [ e^^^ [ eo.^^- ( ea^^+ T e^"^ . J C X JT ^ Jy ^ J '• ^ At points on r, we have 6*«*=0, tf<»<^=l, and so J T X JTX At points on y, 6***= I, so Jy ^ Jy^ Therefore [" ga«fc»<?ff =^(c«-i), /oe J ^oo«6xgin(asin&ar) - =3(e«-l). round the same contour as that used in Examples 7, 8, 9, 10, shew that ('^?^^cLv=Z. jo ^ 2 90 THE PROCESSES OF ANALYSIS. [CHAP. V. Example 12. To find the value of I :r-— cKr, and/ , dx (0<a<l). Jo 1+^ jo 1-^ Write /=/ f— dar,andfi'-/ f—dx. Jo l+a? J Q I -X Ab will be seen from the working below, the integral K has a meaning only when I is + I , where / is a small positive quantity, i-f^ Jo Consider a contour C formed of (a) that half r of a circle, whose centre is at the origin and whose radius is a large quantity 72, which is above the real axis, (b) that half y of a circle whose centre is at the origin and whose radius is a small quantity r, which is above the real axis, (c) that half y' of a circle, whose centre is at the point ( - 1) and whone radius is a small quantity r', which is above the real axis, (d) the parts of the real axis intercepted between these semicircles. — , where the many- valued function is supposed to have that one of its determinations which is real and positive when z is real and positive. The integrand is regular in the interior of the contour (7, and so f 2f-^dz Now on y the integrand is sensibly equal to 2i^'\ and so the integral to — L which -r is infinitesimal, since a > 0. (-1)0-1 On y, the integrand is sensibly equal to ^ J^ — ; putting 1 + « «■ r'tf*', the integral 1 "rZ along y' is / %d$^ or iV ( - 1)«~*. On r, the integrand is sensibly equal to -^^ , the modulus of which is infinitesimal m compared with , -: ; so the integral along F is zero. Thus iri=( -!)*-« /+Jr=-/(cosair-tsincwr) + ir. Therefore equating real and imaginary parts, we have sin air ' K= IT cot air. Example 13. By using the result f'^ x'-^dx _ Jo l+x ~sL sm atr shew that r-^==^—. Limit 2 -r — . (Kronecker.) 68] THE THEORY OF RESIDUES. 91 68. Evaluation of the definite integral of a rational function. The principles which have been applied in the preceding paragraph can also be used to evaluate an integral of the form f f{!c)dx, J —00 where /(a?) is a rational function of x, in the cases when this integral has a meaning. a ioD\ For suppose that f{a) is brought to the form of a quotient Yj-i , where g (x) and h (x) are polynomials in x. In order that the integral may have a meaning unconditionally, it is necessary that the degree of g (x) should be at least two units lower than that of h (x), and that the equation h(x) = should have no real roots. Consider now a contour C, formed of the real axis together with a semicircle F of large radius, whose centre is at the origin, and which lies in the upper half of the z-flsLne. We have 1 f{z)dz — 2iri x sum of residues oi f{z) at the poles oi f{z) J c contained within C, Now / = I + / • *^^ since f{z) has a zero of at least order 2 at jt = 00 , it follows that / is zero. Hence I f{x)dx^ 27n x sum of residues of f{x) at those of its poles / —00 which are contained in the upper half of the ^-plane. If the d^;ree of g (x) is lower than that of A (x) by only one degree, or if A (x) has real non- repeated roots, the integral will still have a meaning provided we make certain restrictions, Le. that I shall be imderstood to mean the limit, when k tends to oo and t to zero^ of where c is a typical root of the equation A (:7) =0. J'-^L' Example I. The function TyTTSs ^^^ ^ single pole in the upper half of the 2-plane, 3i namely at 2»t, and the residue there is -tt.', we have therefore /: dx 3ir (^+1)3 8 * Example 2. Shew that / 7 — — , jr-. = 5-1 • 92 THE PROCESSES OF ANALYSIS. [CHAP. V. 69. GaiLchys integrals We shall next discuss a class of contour-integrals which are very fre- quently found useful in analytical investigations. Let (7 be a contour in the ^-plane, and let f{z) be a function regular everywhere in the interior of C, Let <^ {z) be another function, which in the interior of C has no singularities except poles ; let the zeros of <^ {z) in the interior of C be Oi, Oj, ..., and let their degrees of multiplicity be rj, r„ ...; and let its poles in the interior of (7 be 6i, 6j, ..., and let their degrees of multiplicity be «i, 5a, .... Then by the fundamental theorem on residues, we have ^ — ; I f(z) %-T-l dz = sum of residues of -4^-}—- in the interior of C. 2'inJc <t>{^) <f>(^) Now !T^ c*^ bave singularities only at the poles and zeros of <f> (z). At one of the zeros, say Oi, we have <f>{z) = A(z- OiY' +B(z-aiy^+' + .... Therefore <^' (z) = Ar^ (z - o,)'"*-^ + B(n + l){Z''a,y^-{- ..., and /(z) =/(a,) + (^ - Oi)/' (aj) + . . . . Therefore !f^ = + a' constant + positive powers of (z — Oi). <P yZ) Z — Oi Thus the residue of !T} > at the point ^ = Oi , is r^f{a^. Similarly the residue at -er = 6i is — «i/(6i) ; for near «r =s 6^, we have and f{z) =/{bO + (-? - 6,)/' (bi) + • • ^ f(z)6' (z) -5,/(6i) . . r r so "^ ^ , ; . a= I + a constant + positive powers of jp — 6i. <f>{z) z-bi ^ ^ ' Hence ^ f^/{z) *'|^J dz^t rj(a,) - 2 .,/(60, the summations being extended over all the zeros and poles of <f> (z), 60. The number of roots of an equation contained within a contour. The result of the preceding paragraph can be at once applied to find the number of roots of an equation <f>{z) = contained within a contour C, For on putting f{z) = 1 in the preceding result, we obtain the result that ^T— ^ I ^"7 \ dz is equal to the excess of the number of zeros over the number 2mJc<f>{^) 59 — 61] THE THEORY OF RESIDUES. 93 of poles of if>(z) contained in the interior of (7, each pole and zero being reckoned according to its degree of multiplicity. Example 1. Shew that a polynomial (t) of degree m has m roots. Let <^(«) = ao2^+ai«^"* + ...+am- Then ^ = ^og;r^^ >"-+^- ^ . "" _ * y2' " " For large values of «, this can be expanded in the form *'W = ^ + ^ <l>{z) z^z^ Thus if C be a large circle whose centre is at the origin, we have Hence as ^ (z) has no poles in the interior of C, we have number of zeros of d> (z) = ^j— : j %-rl cfe" m. Example 2. If at all points of a contour C the inequality is satisfied, then the contour contains k roots of the equation For write /(«)=a^a!~+a„»--i«^~^+...+ai«+aQ. Then /(«)=«».» A^ a^^-»+...+«.,, j^+ a^.^-'+...+a, ^ =a*«* (1 + i7) say, where | CT"! < 1 on the contour. Therefore the niunber of roots of f{z) contained in C 2irt ; c /W "2irijcV« 1 + C^ rfW — =27rt ; and since |^| < 1 we can expand (1+ (7)~* in the form Therefore the number of roots contained in C is equal to k. 61. Connexion between the zeros of a function and the zeros ofUe derivate, Macdonald* has shewn that if f(z) be a regular function of z in the interior of a contour C, defined by an equation \f{z)\=M where M is a constant, then the number of zeros of f(z) in this region exceeds the number of zeros of the derived function f'{z) in the same region by unity, * Proc. Land, Math. Soc, xzix. (1898). 94 THE PROCESSES OF ANALYSIS. [CHAP. V, For since f{i) has no essential singularity in the region, the number N of its zeros in the region is finite. Now if m be a small number, the part of the locus |/(«)|=m in the interior of the contour C consists of N closed curves surrounding the N zeros of f{z). As m increases, these ovals increase, until two of them coalesce, the point at which they coalesce being a node on the curve corresponding to that particular value of m. When m has increased to its final value if, the N closed curves have coalesced into one closed curve, and therefore N^-l nodes have been passed through. Each of these nodes is a zero of f (z); for if /(«)=0+«V^, where and yfr are functions of x and y with real coefficients, then ^ and -^ vanish at a node on the curve ^+^= constant; that is, f'{z) vanishes. Moreover, two ovals cannot coalesce at more than one point, b» f{z) is single-valued. Hence the number of zeros of /' (z) inside the contour is (^- 1). The proof assumes the zeros of f(z) in the interior of (7 to be all simple : the case where f{z) has multiple zeros can be at once reduced to this, by dividing out the factor common to f{z) €Uid /' (z). If /' (z) has two zeros equal, two of the double points coalesce, that is, three ovals coalesce at the same point. Similarly it can be shewn that the number of zeros of /' {z) in the region between the contours |/(«)|«=mi and |/(«)|"««ij is equal to the number of zeros of f(z) in the same region, if /(«) is regular in the region. Example 1. Deduce from Macdonald's result the theorem that a polynomial of degree n has n zeros. Example 2. Deduce from Macdonold's result that if a ftmction /(z), regular for real finite values of z, has all its coefficients real, and all its zeros real and different, then between two consecutive zeros of f(z) there is one zero and one only of f'{z). Miscellaneous Examples. 1. a function 0(«) is zero for «=0 and regular when I^Kl. If /(x, y) is the coefficient of » in <^ (^+yO> prove that 'o i^2a>>cos^-h ^ -^(^^> sm^)cW-W>W. (Trinity College Examination, 1898.) « «i. XI- X r* sinew; , 1 e«+l 1 /t j v 2. Shew that j^ -__<ir-j^y-y--. (Legendre.) 3. By integrating I e-^dz round the perimeter of a rectangle of which one side is the real axis and another side is parallel to the real axis and at a distance a from it, shew that /:. e-'*cos ^aidt^sfne-^\ '00 and I e-^ Bin 2(Udt^0, /: oi- XL X [ l-rco8 2^ , . ^j^ w, \-r 4. Shew that / , — ^^-r^ log sm 6dB = t log —j- jo l-2rcos2^+r* ^ 4 ® 4 MISC. EXS.] THE THEORY OF RESIDUES. 95 6. Shew that /, a?cwr=--log(l+a) if -.l<a<l l-2aco6a?+a* 4a and = ^ log ^1 + ^^ if a« > 1. (Cauchy.) 6. Shew that /, 00 sin 01^ sin (bqX sin d>-^ sin o^ , ir . . • •• if a be different from zero and (Stermer.) 7. If a point z describes a circle C of centre a, any one- valued function u^f{z) will describe a closed curve y in the t«-plane. Shew that if to each element of y be attributed a mass proportional to the corresponding element of (7, the centre of gravity of y is the f{z) point r, where r is the sum of the residues of -^^-^ at poles in the interior of C. z — a (Amigues.) 8. Shew that dx 7r(2q-hft) i-«(^ -00 (^+ 6^) (^ + a2)« 2a26 {a-^hf ' 9. Shew that dx _ TT 1 .3. ..(2^-3) 1 / »oo(a+6j?2)* 2*-»6* 1.2...(n-l) a^-h' 10. If /VW=(l-^)(l-47«)...(l-a;*-i)...(l-ar«)(l--^)... (1-;f«*-%. ... (l-a?»-i)(l-:c«»-«)... (l-x(»»-»>"), shew that the series convergee when »■ is not a root of one of the equations 1-0; ©■- and that the sum of the residues of f{x) contained in the ring-shaped space included between two circles whose centres are at the origin, one having a small radius and the other having a radius between n and n-|-l) is equal to the number of prime numbers less thann-hl. (Laurent.) CHAPTER VL The Expansion of Functions in Infinite Series. 62. Darbov^'s formula. Darboux has given* a formula from which a large number of expansions in infinite series can be derived Let f{z) be an analytic function of z, regulai* at all points z within a circle of centre a and radius r; and let z he a, point within this circle. Let (z) be any polynomial in z, of degree w. Then if R^ denotes the expression (- 1)« {z - a)*»+' C <f> (0/<"-^*' {a + t{z^ a)] dt, Jo where the integration is taken along the real axis of t, we have on integration by parts R^ = [\- ir{z- aY <f> {t)f^^ {a + e (^ - a)}J + (- 1)«-^ (z - a)« f'f (O/^*** {a + t(z-- a)} dt, J Q or i2n = (- lY (^ - ar {<t> (!)/<"» (z) - <t> (0)/<«» (a)} + (- D^^i {z - aY fV' (0/*"^ [a + t(z^ a)} dt Jo Integrating the last integral by parts in the same way, we obtain i2„= (- ir (z - ar {<!> (l)/<~> W - <t> (0)/<«) (a)} + (- 1)~-» (^ - a)*^^ {f (I)/<'^»> (-?) - f (0)/(«-« (a)) + ... - (^ - a) (<^<«-^) (1) r (^) - <^'^^^ (0)/' (a)} + (-e - a) f <^ w (0/' [a+tiz"- a)] dt. Jo Now <^<*> (t) is a constant independent of t, since <^ (t) is a polynomial of order n ; and hence (z - a) f ' <^^«) (0/' {a + < (^ - a)} dt = <^» (0) {/(z) -/(a)}. ♦ Liouville'8 Journal (3), u. (1876), p. 271. 62, 68] THE EXPANSION OP FUNCTIONS IN INFINITE SERIES. 97 Thus finally we have Darboux's formula <f>^^ (0) {f(z) -/(a)} ^(z^a) {<^*-« (1)/' (z) - <^^-« (0)/' (a)]... + (- 1)» (iT - a)^' r (f> (t)/^"^'^ {a + t(z- a)} dt Jo Taylor's expansion may be derived from this formula by putting ^(^)aB(t — 1)", and then making n tend to infinity: other new expansions may be obtained by substituting special polynomials of degree n for <f> (t), and in the resulting formula making n tend to infinity : in each case it must of course be shewn that Rn tends to zero as n tends to infinity. Example, By subetituting 2n for n in Darbouz's formula, and taking it>{t)=:t^(t- 1)% obtain the expansion /(«)-/(«)- J^ ^"^^ynr''^V <"W+(-l)'"-V<"(a)}, and find the expreesion for the remaioder after n terms in this series. 63. The BemoulUan numbers and the Bemotdlian polynomials. • z z If the constants which occur in the expansion of ^ cot ^ fin ascending powers of jt be denoted by J3i, jBs, J3t» ... > so that z z s^ * s^ ^ then Bn is called the nth BemoulUan number. It is found that -''1 — g I -^^a — 80 » •'^t = 42 > • • • • The BemoulUan numbers can be expressed as definite integrals in the following way. TTT 1 r* sin ©a? da? S f* . , We have I J; , = 2 / €r^^ sm pxdx 00 = 2 — *^ — = -1+1 Equating coefficients of j)*'*~' on the two sides of this equation, and writing a = 2t, we obtain A proof of this result, depending on contour integration, is given by Carda, MoncUthefte fwr Math, und Phys. v. (1894), pp. 321-4 W. A. ' 7 98 THE PROCESSES OF ANALYSIS. [CHAP. VI. BxampU. Shew that * ir**(2»»-l)yo "smhF* The Bemoullian polynomial of order n is defined to be the coefficient of t* e** — 1 . —. in the expansion of t -^rZTT ^ ascending powers of t It is denoted by <f>n (z), 80 that eV-T= 2 ^V (!)• 6*— 1 »■! n! ^ -^ This function possesses several important properties. Writing (z + 1) for z in the preceding equation and taking the difiference of the two results, we have n-1 Tli On equating coefficients of t^ on both sides of this equation we obtain nz'^^ = <f>n(z+l)-<f>n(z\ which is a difference-equation satisfied by the function ^^ (z). The explicit expression of the Bemoullian polynomials can be obtained as follows. We have and e*- t tef + 1 t «*- l~2c«-l 2 ~2 £ 4 2 = 2i~*2i-2 "^"2"^ 21 ~ 4! ■^•••• Hence „r,~in r"^ 2r"^"3r'*'-|f "2■'"■2^"^■^••r From this, by equating coefficients of <*, we have the last term being that in ^ or ^ ; this is the explicit expression of the nth Bemoullian polynomial. mm 64] THE EXPANSION OF FUNCTIONS IN 'INFINITE SERIES. 99 The BemouUian numbers and polTnomials were introduced into analysis by Jacob Bernoulli in 1713. Example. Shew that *nW=(-ir*i»(i-«). 64. The Maclaurin-Bernoullian expansion. In Darboux's formula write <^ {t) = ^n (0> where 4>n (0 '^ ^^^ ^^^^ Bemoul- lian polynomial Now from the equation <^(^+l)-.<^„(0 = n^-S we have by differentiating k times <^n«*»(e + l)-i^n^(0 = w(n-l)...(n-A)r-*-^ Putting ^ s in this, we have But the value of <^n^' (0) is obtained by comparing the expansion <^(^) = 0n(O) + ^<^,'(O)+|j<^"(O) + ... with the expansion Substituting the values of <^„**(1) and ^n**(0) thus obtained in Darboux's result, we find what is known as the Ma^^laurin-Bernoullianformtda, (z - a)/' (a) ^f{z) -/{a) - —^ {/' (,z) -/' (a)} + ^i%P^{/"(^)-/"(a)l + -. •^ ^" ^^'(In"- 2)7 "^ t/*^ (^) -/"^ («)} In certain cases the last term tends to zero as n tends to infinity, and we can thus derive an infinite series from the formula. Example, lff{z) be an odd function of «, shew that JV*.(<)/«***"(-«+2«)<ft, 2nl where ^. (t) is the Bemoullian polynomial of order n. 7—2 / 100 THE PROCESSES OF ANALYSIS. [CHAP. VI. 66. Burmann's theorem. We shall next consider a number of theorems which have for their object ths expansion of one function in powers of another function. Let <f> (z) be a function of z, which takes the value b when z takes the value a, so that b = <t> (a). Suppose that ^ (z) is an analytic function of z^ regular in the neighbour- hood of the value -e? = a, and that (f/ (a) is not zero. Then Taylor's theorem furnishes the expansion <l>iz)-b = 4>'(a)(z-a) + ^^iz-ay+..., and on reversing this series we obtain which expresses -^ as a regular function of the variable {<f>{z) — 6}, for values of z in the neighbourhood of a. If then f(z) be a regular function of ^ in the neighbourhood of a, it follows therefore that /(-«?) is a regular function of {<l> (z) — 6} in this neighbourhood, and so an expansion of the form A') =/(«) + «! {* (^) - 6} + Jj {4> (^) - b}* will exist, which, as it is a power-series in {^ (z) — 6}, will be valid so long as \^(z)-b\<r, where r is some constant. The actual expansion is given by the following theorem, which is generally known as Burmann's theorem. If yfr (z) be a function of z defined by the equation , . z — a then the function f(z) can for a certain domain of valines of z be ea^pa/nded in the form fiz) =/(a) + 1^ ^^^^i,"^^" ^. [/' (a) tt («)}-] ; and the remainder after n terms in the series is i_ c' f r <i,(z)-b -\'-^f'(t )<f>'(z)dtdz 2',rijjyl4>{t)-bj if>(t)-<l>(z) where y is a simple contour in the t-plane, enclosing the point < = a. 66] THE EXPANSION OP FUNCTIONS IN INFINITE SERIES. 101 To prove this, we have - A. ['[ /' (0 <t>' (^) dt dz r ^(z)-h ^ iinJJy <f>{t)-b I TW^ {^(^)-fel-' -l But —['[ \^(_')-i'\ V(t)<l>'(')cUdz {»(^)-6}*-» f /'(t)dt ZmJaJyl'l>(t)-bj (f>{t)-b 2in{k+l) Jy{<t>it)-b}'^' 2^- (A + 1) J^ (< _ a)*+' - 2,ri(A+l)! ^* ^-^ («> t^ ^">5 J' Therefore /(^)=/(a)/T^ ^^ii^* £.[/'(«) ftW] Exampli Lf'f lt(ft:lT~'/l(tl£Mdtdz '^JaJyl<l>{t)-bj <f>{t)-<f>(2) • where 27rt To obtain this expansion, write in the above expression of Burmann's theore^n ; we thus have Zi But (&J '"^•'"•^}« -{^1 ''''^"'^),,, (P^**i^« *=«+') =(n-l) ! X coefficient of ^-i in the expansion of tf-«^(>«+<) =(n-I)!xooefficientof^-iin i (-l)^^^<^(g«+<r r-o rl ^ ^ A(n-l-r)!(2r-n+l)r The highest value of r which gives a term in the summation is r=n- 1. Arranging therefore the summation in descending indices r, beginning with r«n- 1, we have =(-i)«-ic;. which gives the required result «•« ip 102 THE PROCESSES OF ANALTSia [chap. VI. Excmiple 2. Obtain the expression ««-sin««+- . 2®^ '+3~5 • 3 8m««+ .... Example 3. Let a line p be drawn through the origin in the ;s-plane, perpendicular to the line which joins the origin to any point a. It zh% any point on the «-plane which is on the same side of the line p as the point a is, shew that 00 1 /«—a\ '"*■•' ^ 66. Teixeira's extended form of Bunnann's theorem. In the last paragraph we have not investigated closely the conditions of convergence of Burmann's series, for the reason that the theorem itself will next be stated in a much more general form, which bears the same relation to the theorem just given that Laurent's theorem bears to Taylors series: viz., in the last paragraph we were concerned only with the expansion of a function in positive powers of another function, whereas we shall now discuss the expansion of a function in positive and negative powers of the second function. The general statement of the theorem is due to Teixeira*, whose exposi- tion we shall follow in the next two paragraphs. Suppose (1) that/(^) is a regular function of ^ in a ring-shaped region Ay boundid by an outer curve S and an inner curve s; (2) that 0{z) ia a regular function everywhere inside /S, and has a single zero a within this contour; (3) that x is the affix of some point within A; (4) that for all points of the contour S we have l^(a;)|< 1^(^)1, and for all points of the contour s we have i^(a;)i>i^(^)i. The equation has, in this case, a single root z=»x in. the interior of S, as is seen from the equation 1 r e'(z)dz _ I [[ ff{z) , ^ ., . r ff{z) i ^ J_ r ff(z)dz ^27riJs 0{z) ' of which the left-hand and right-hand members represent respectively the number of roots of the equation considered and that of the roots of the equation 6{z)rsQ contained within 8. * CreUe'9 Journal, cxxn. (1900), pp. 97—123. mmm 66, 67] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 103 Cauchy's theorem therefore gives /M = J_ [( A^W('')ds [ Az)0'{z)dz -\ ^^*^ 2%'7rlJae{z)-e(x) J, '^(z)-0(x)]• The integrals in this formula can, as in Laurent's theorem, be expanded in powers of 0(ai), by the formulae f f{z)ff{z)dz _lff,,.t Az)ff{z)dz We thus have the formula where a ^ JL { fMli^l^ Sn=i^\nz)e^^{z)ff{z)dz. This gives a development oi f{x) according to positive and negative powers of 6 {x\ valid for all points x within the ring-shaped space A. 67. Evaluation of the coefficients. If the function /(^) has no singularities but poles in the region limited by the curve a, the integrals which occur in the preceding formula can be evaluated in the following way. Let 6i, &9, ... &A be the poles; and let Ci, C2, ... c*, c, be circles with centres &i, 6s, ... 6;^, a, respectively, and with very small radiL Then A - ^ ( A^)^(')d z_ 1 f /'(z)dz "'iiriJs 0^' {z) ~ liri } s ne^ (,zj = S JL/" f'i^ldz 1 (f'{z)dz and Bn'^'-^j f{z)e^'{z)ff(z)dz a?— a Ix^a 104 THE PROCBSSBS OF ANALTS1& [CHAP. VI. Thus if Om be the degree of multiplicity of the pole 6,», and if ^ ^ be denoted by 0i (x), we have " n!L<iB»-'K»(«)jJ, + I 1 r<^- y'(a>)(a>-6m)-^M '| It may happen that a is also a pole of /(x). It is easily seen that in this case A^, is given by the formula ■^ (n + /9) ! n [dafi+^ \ ^,» (a;] where /3 is the degree of multiplicity of the pole a ; the formula for Bn must likewise be replaced by - ^inoni [^ ^f <*> ^'" <-) <- - '^>'"-"j].^' when w ^ )8. The preceding formulae do not give the value of Aq ; this can be found from the formula A ^ i l_f f{^)^i^)dz . 1 { f{z)ff{z)dz "^'"jLiiiTrJc^ 0iz) ^iiirj, 0(z) ' which gives when a is a regular point {oTf{x) ; and . l[d? \ f{x)ff{x){x-af \-[ when a is a pole of /(as). 68] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 105 Example 1. Shew that when — 1 < ^ < 1. ^"sVr+^j'^iTid+s) "^ 27176 (i^-^^ "*■•••' Shew that the seoond member represents - , when | a? | > 1. Example 2. If S^^ denote the sum of all combinations of the numbers 2«, 4«, 6«,...(2n-2)«, taken m together, shew that « sin«^^(2n-|-2)! l2n-|-3 2n-|-l ^-^ 3 p®"^^^ ' the expansion being valid for all values of z represented by points within the oval whose equation is | sin f | « 1 and which contains the point ;?=0. (Teixeira.) 68. Expansion of a function of a root of an equation, in terms of a parameter occurring in the equation. Now consider the equation 0{x)^(x-a)0i{x) = t, where ^ is a number such that along the contour 8 we have |^ (^)| > |^|> and along the contour s we have \0(z)\ < \t\. The equation (x) » t, regarded as an equation in x, will then have a single root in the ring-shaped region bounded by the curves S and s; we see, in £Btct, from the equations 2m}a0W^t-2mlJsW) Js^) ^'"i -1, -0, that the equation in question has one root in the interior of S and none in the interior of s. Then if the function /(x) is regular in the region limited by 8 and «, we see from the preceding articles that the formula where An and Bn have the values already found, gives the expansion in powers of t of the function /(a?) of the root considered. As an example of this formula consider the equation {x - a) oosec x^^t, and let f(-r)=-i-. ^ ' x-a 106 THE PROCESSES OF ANALYSIS. [CHAP. VL Then we find . cosa " Sin a' *" (n+l)ln ^-+1 ' * sin a' * ' Hence cosa * t^ cP*+^(8in*a) . 1 x-a sin a n-i(w + l)In cto»*i <sina' and thus gives the expansion, in ascending powers of t, of , where x is given in terms of t by the equation 0? = a + < sin ^. (Teixeira. ) 69. Lagrange's theorem. Suppose now that the function /(^) is regular at all points in the interior of £•, 80 that the poles 6n &s, ... &a do not exist. Then the formulae which give the quantities An and 5« now become 1 d*-» m <"-'• ^0 =/(a), Moreover the contour s can now be dispensed with, and the theorem of the last article takes the following form : Let /(z) be a regular function of z at all points in the interior of a contour fif, and let 6 (z) be a regular function with no zero in the interior of S. Let a be a point inside iS, and t a number such that for all points z on S wo have \(z-a)e(z)\>\t\. Then the equation (z — a)0 (z) = t will have one root x in the interior of S, and /(a;) will be given as a power-series in t by the expansion /(x)^f(a) + l -,J^ «— 1 r. e^(ax This result was published by Lagrange in 1768 ; it is usually stated in a slightly different form, to obtain which we shall write the result may now be enunciated as follows : ^//(^) ^^ ^ W ^ regular functions of z within a contour S surrounding a point a, and if the a quantity such thai the inequality t<l>{z)\<\z-a 69] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 107 18 satisfied at all points z on the perimeter of £>, then the equation z=sa + t<l>(z\ regarded as an equation in z, has one root in tlie interior of S: and if this root be denoted by a?, then any regular function of x can be expanded as a power- series in t by the formula This result is of course a particular case of the more general theorem given in § 68. Example 1. Within the contour surrounding ;e—a and defined hj the inequality |;?(«-a)|>|a|, the equation «-a— -=0 has one root z, the expansion of which is given by Lagrange's theorem in the form Now jfrom the ordinary theory of quadratic equations, we know that the equation z—a — =0 z has two roots, namely and our expansion represents the former of these ordy — an example of the need for care in the discussion of these series. If however we regard the expansion as a power-series in a, and derive other power-series from it by continuation in the a-plane, we shall ultimately arrive at the series « (-l)»(27i~l)! g'* »Ii n\ (n-l)I a*»-i' which represents the other branch of the function z. Example 2. If y be that one of the roots of the equation y^\-\-zf which reduces to unity when z is zero, shew that »(»+6)(w+6)(»+7) ^ w(w+6)(»+7)(»+8)(»+9) ^ ^ A. \ n \ 41 ■ 6! so long as \z\<^. Example 3. If ^ be that one of the roots of the equation 108 THE PROCESSES OF ANALYSIS. [CHAP. VI. which reduces to unity when y is zero, shew that the expansion being valid so long as \y\ < |(a-l)«-ia-«|. (McClintock.) 70. ItoucM*s extension of Lagrange^ a iheorem. Consider now two functions /(^) and {z), which are regular at all points within a contour 0, on the perimeter of which the inequality j j;.\^ A") satisfied. < 1 is Then we shall shew that if the equation f(z)=:0 have p roots a^y a^, ... Op in the region contained by C, the equation f{z) — a<f> (z) = tuUl have p roots Oi', Oa', .•• ^'i *w ^f^ region; and for every function F(z) regular in the region we shall have r=l r=l n=l ^' r«l aO^ I -^^ (0.^)] ) f(z) where ylr (z) = ^ ' . We may note that this theorem reduces to that of Lagrange when f(z) ^z^a and p = 1. The result stated may be obtained in the following way : We have 2 J'(a,')-K^. f ^'/-^M^^^^dz {/(^)}'{/(^)-«<A(^)}J t/wl /(^) 1/(4 / (^) - «<^ w J ' When n is large, the last integral tends to zero: we thus have on the right-hand side a power-aeries in a, in which the coefficient of a* is or 4 i r j^' \ F' (z) {<!> (z)]' jz - a,)"> [l r> ^ ■ ■ ■ ; I .'' J. t w m _^„ ^s: , ^j.i., . 1 j e^i^^mf^^m 70—72] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 10& or I i JUl \ r (Or) 4> (ar)n which establishes the theorem. Putting F{z) = 1, it is seen that the number of roots a' is p. 71. Teixeira has published the following generalisation of Lagrange's theorem, the proof of which may be left to the student. Let where ^ (z)^ ..,<t>k (z) are regular functions of z in the interior of a contour K, and ^ is a point inside K, Let a be a positive quantity, so small that the condition z-t z-t + ...+ a*0fc {z) z-t <1 is satisfied along the contour K, Then to every value of x which satisfies the condition I J? I < a there corresponds a unique value of « in the interior of K ; and / («), where / is a regular function at all points in the interior of JT, can be expanded in ascending powers of X by the formula where the summation is extended over all positive integral solutions of the equation a + 2^+3y+...+i{-X«n, and where Another form of this result is /W-/W +22 7-^ ^, {/' W *r-,. M (0}, M-K> v-0 syT*-) ' dt where the quantities <^v,^ are obtained from the equations 72. Laplace's extension of Lagrange's theorem, Lagrange's result can easily be extended to a case in which the given equation is of a somewhat more general type. Suppose that the equation is given, and that it is desired to expand some function f{z) of a root of this equation in ascending powers of t If we write a + ^0 {z) = u, the equation reduces to i^ = a + <<^ {i/r (t^)}. The problem of expanding /(^) is therefore equivalent to that of expand- ing/{'^(w)}, where u is given by the last equation; and this can be done by Lagrange's theorem. 110 THE PROCESSES OF ANALYSIS. [CUAP. YI. 73, A further generoMsaiion of TayMs theorem. The series of Laurent, Darbouz, Burmann, etc. may be regarded as extensions in different directions of the fundamental series of Taylor. A generalisation of Taylor's theorem of a somewhat different character to these, is furnished by the foUowing result, the proof of which may be left to the student. Iff{z) and B {£) are regvlar functione oft in the neighbourhood of the point zt^x, and if 6, ('^)^jj (t) dt, 3, W=//i W dt, and generally 3n(^)=j\^iWdt, then, for values of tin the neighbourhood of the point x^ f{t) can be expanded in a series of the form /(«)«=ao^W+MiW+«2^2(«)+.-+«»^« («)+...> where and generally the number of differentiations in the last expression being n. It is clear that Taylor's series is obtained from this expansion by putting B {t)=h Example 1. Shew that Example 2. Shew that (Laurent, Joum. Math. Sp4c., 1897.) Example 3. By writing B {t)^e^, obtain the expansion of an arbitrary function of f in a series of the form where o^^, o^ are independent of t. Example 4. In the general result, shew that when ;r=0 we have where /W-2~5^ and d(«)-2^«*. ^i^and d(^)-2^, ni ^ ' n\ (Quichard, Annates de Vic. Norm,, 1887.) 78, 74] THE EXPANSION OP FUNCTIONS IN INFINITE SERIES. Ill 74. The expansion of a function in rational functions. Consider now a function f(z), whose only singularities in the finite part of the plane are simple poles Oi, a,, Os, ...: let C], c^, ... be the residues at these poles, and let (7 be a circle of very large radius R not passing through any poles, so that f{z) is finite at all points in the circumference of C. (The function cosec^ may be cited as an example of the class of functions con- sidered.) Suppose further that at all points on the circumference of (7, the modulus of f(z) is less than M, where if is a quantity which remains finite when large values of R are taken. Then jr—, I J-^ dz^snm of residues of "^^ at points in the interior 2w% JoZ — ic z -X ^ of (7 =/(a?) + 2-^^, . where the summation extends over all poles in the interior of C. But ±( f<^dz = —(^-^^ + —.l -^^^dz iiriJce — m iirijo " 2in ] a z {z — w) -^ ^ ^ ^an 2inJoZ{Z'-x) if we suppose the function/(^) to be regular at the origin. Now R being supposed large, I -j^^—k is of the order -^ of small quantities, J gZyZ ^^X) JX and so tends to zero as R tends to infinity. Therefore on making R infinitely great, we have = /(.)-/(0) + 2c„(^-l), or m.m^%^U^*i\. which is an expansion off{x) in rational functions of ^. If instead of thd condition |/(«)| < J^ we have the condition \f{z) \ < MB^, where J^is finite for all values of R and n is a positive int^er, then we should have to ezpemd F{z)dz I o z—x by writing cosec;?: and should obtain a similar*but somewhat more complicated expansion. Example I. Prove that z ^ * \z-7iir nirj the summation extending to all positive and negative values of n. To obtain this result, let cosec z — =/(2)* The singularities of this function are at the z points issn^, where n is any positive or negative integer. ■■■■JHW jjiwii j^.^^Mn^^PHPK^mnm ^ 112 THE PROCESSES OF ANALYSIS. [CHAP. VI. For points near one of these singularities, put ^smr + (1 Then /W=cosec(wir+f) l-^ilil^ ^ J- (l^-LY^ ^ h positive powers of (. z — nir nir The residue of /(«) at the singularity nir is therefore ( - 1)*. Applying now the general theorem /(.)=/(0)+.c,[^ + I]. where c^^ is the residue at the singularity a^, we have /«=/(o)+s(-i)-j^-L_+i^}. But /(O) «» Lt,.o - + (positive powers of «) - - =0. Therefore cosecfs which is the required result. Example 2. If a is real and positive and less than unity, shew that 6^ 1 « 2« cos 2nair - 4nir sin 2nan' For ^f{z)^- — r — , the singularities of f(z) are at the points zs^^nni, where n«=±l, ±2, ±3, ... ±00. For points z near z « 2wirt, put z = 2nirt + f . Then = — — +a series of positive powers of f. The residue at z^s^nin is therefore e*^'*. Also I • /(0) = fl±5±::.-n = [i(l+«.+ ...)(l+|+...)"-i2_^ Applying the general theorem /W=/(0)+2c,(^ + l). we have therefore H±«) ^i»a 00 sin2natr 2 — T-^+ 2 EXS.] THE EXPANSION OF FUNCmONS IN INFINITE SERIES. 113 But - 2^. log ( - «-»^) = ^. (irt - 2a,r0 1 Thus «• — 1 z N.±i # — 27ii7r " «-i \« - 2mV z + 2niir) "* 2^ COB 2nafl' — 4n9r sin ^nair Example 3. Prove that 1 ^_1_ _ 1_ I _2 1 7r«*(fl«-2co8^+«~*)"2fra?* «*-«-* ir* + Jar* «*»-«-«» (2ir)*+i^ For the general term of the series on the right is {-\Yr _ 1__ which is the residue at either of the four singularities r, -ryvi^ -ri^ot the function vz (ir*a^ - JjF*) {eF* - e ~ »*) sin irz ' The singularities of this latter function which are not of the type r, -r, f% —riy are at the points ^ ±s/i X ± sT-^i X V2 "• \/2 «■ 2 At «a=0 the residue is -^; at either of the four points «= ~ J" — , the residue is v2 'f' ir«~l /Vife _>/jr ^ ) . VtJr S/ sin ._ V2 Therefore __ 1 /" ffgcfe ~ 2wt j c ('T*'?* - J^) (e^* - e - »«) sin irz * where C is an infinite contour. But at points on C, this int^prand is infinitely small compared with - ; the integral round C is therefore zero. W. A. 8 114 THE PROCESSES OF ANALYSIS. [CHAP. VI. 1 - (-lyr 1 :ii -1 ,^(;--)^_«'-"l}{.'-'>l _,»-«!} -1 1 ira^ (e« - 2 cos ^+«~«) ' which is the required result. Example 4. Prove that Example 5. Prove that coseoh*=l - Ste (^ - ^;^ + gj^ ...) . Example 6. Prove that 8ech;r=4ir(^^,^^-^^^ + 2j^a^^...j. Example 7. Prove that coth *-l + to (^ + ^^-, + g^+ . ..) . Example 8. Prove that 2 2 7—5- — o> , g . ,gv = -7 coth ira coth irb. (Cambridge Mathematical Tripos, Part I, 1899.) 76. Expansion of a function in an infinite product The theorem of the last article can be applied to the expansion of functions as infinite products. For let / (z) be a function, which has simple zeros at the points Oi , a,, 0,, . . . where Limit | On | is infinite ; and suppose that f{z) has no singularities in the finite part of the plane. Then clearly /' (z) can have no singularities in the finite part of the f (z) plane, and 80*^-77^ can have singularities only at the places Oi, a,, a,, .... 75] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 115 Now for values o(z near Or, we have by Taylor's theorem /(z) = (^ - ar)f (a,) H- (l^f' (a,) + . . . and /' (^) =/' (a,) + (z^ Or)/" («r) + • • • • Thus we have :r7\ ~ ^ * constant + positive powers of (z — Or). J yZ) Z — Or f'(z) At each of the points Or, the function ''-^it^ has therefore a simple pole, with the residue + 1. f (z) . . If then -zj-r has at infinity the character of the functions considered in the last theorem, it can be expanded in the form Integrating this expression, and raising it to the exponential, we have where c is a constant independent of z. Putting 5 ~ 0, we see that/(0) = c, and thus the general result becomes f'io) This furnishes the expansion, in the form of an infinite product, of any function /(z) which fulfils the conditions stated. This theorem is a case of a general theorem on the factorisation of functions, which is due to Weierstrass, and which will be found in Forsjrth's Theory of Functions, Chapter v. Example 1. Consider the function /(«)< z , which has simple zeros at the points rn-, where r is any positive or negative integer. In this case we have /(O) =« 1 , /' (0) = 0, and so the theorem gives immediately sin ^'-.?«{(-4)-l' since the condition relative to the behaviour of "^tt-t at infinity is easily seen to be fulfilled. 8—2 116 THE PROCESSES OF ANALTSia [CUAP. VI. Example 2. Prove that h©} {-(s.!.)} {-(s^.)} {-U^j} (-(^^1 cosh k - COB X 1 — oosx (Trinity College ExaminatioD, 1899.) 76. Expansion of a periodic function in a series of cotangents. Another mode of expansion, which may be applied to periodic functions whose poles are all simple, is that indicated in the following example. Consider the function cot {x — Oi) cot (a: — a,) ... cot (x — On). This is a trigonometric function of a?, having poles at the points Oi, a,, ... On, and also at all other points whose afiSxes differ from one of these quantities by a multiple of tt. There is clearly no loss of generality in supposing that the real part of each of the quantities Oi, a,, ... 0^, lies between and tt. Now let ABCD be a rectangle in the j^-plane whose comers are the points J[(^ = — too), £(^sB7r— 100 ), C(j?=7r-»- too), and D(^ = too); and consider the integral ^r — A cot (if — Oi) cot (if — Oj) ... cot (-^ — On) cot (j? — a?) ck taken round the perimeter of the rectangle. The integrals along DA and CB are equal but of opposite sign and cancel each other. Along CD, each of the cotangents has the value - 1, so the integral along CD is . Similarly the integral along AB has the value •5- . The whole integral has therefore the value l+(-l)\., %\ 2 The singularities of the integrand in the interior of the contour are at the points 2^ = Oi, Os, ... On, X ; and clearly the residue at tir is cot(ar — cti)cot(ar — aa)...cot(a,. — ar-i)cot(ar — Or+i) ••• cot (ttr — On) cot (Oy — x), while the residue at a; is cot(a: — Oi) ... cot(a? — ttn). Since the value of the integral is equal to the sum of all these residues, we thus have 1 4.(- l)n • r=i» ^— ^ t** = cot(a? — cti) ... cot (a? — On) + 2 cot(a,. — Oi) ... cot {Obr — a>f^ cot (Or — x). 76, 77] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 117 Thus if n be even, we have cot (a: — Oi) ... cot(a? — On) = 2 cot(ar — ch) • ••cot (or — On) cot (a? — Ctr) + (—l)*, and if n be odd we have cot(d? — Oi) ... cot(a?-a„)= 2 cot (ctr — cti) . . . cot (cv — On) cot (a? — a^). This method of decomposition into a series of cotangents is of very general application to periodic functions ; it may be regarded as the trigouo- metrical analogue of the decomposition of a rational function into partial fractions. Example. Prove that 8m( 3? - ^i) sin (a? - fcj) • • • ^^^ (^ ~ K) ^^ (<*i "" ^i) • • • ^^^ (*i " ^1*) Bin (47 -a^) sin (a? -Oj) ...8in(a:-a„) sin (aj-Oj)... sin (aj-oj cot(4;— Oj) 8m(aj-aj)...S]n(a2-aJ ^ + + C06 (a|+a2+...+a„-fci — 6j— ... — 6i»). 77. Expansion in invei^se factorials. Another mode of development of functions, which although investigated by Schlomilch as long ago as 1863 has hitherto not been much used*, is that of expansion in inverse factorials. Let Z be a line drawn parallel to the imaginary axis in the j^-plane ; and draw a circle of large radius, having its centre at the point where I cuts the real axis. Consider a function f{z\ which has no singularities within the semi- circular area which is bounded by I and this circle and which lies on the positive side of Z ; let 7 be the semi-circular arc which bounds this region. Suppose moreover that at all points of 7 we have the inequality \f{z) i < M satisfied, where M is finite however large the radius of 7 may be chosen. Then if z be a point within this semi-circular region, we have Now [ f(t)dt ^r f(t)dt r zf(t) dt ♦ Beferenoes to some recent work are given by Klajver, CampUs Rendui, oxxxir. (1902), p. 687. ^^1 — - ^> K* 118 THE PROCESSES OF ANALYSIS. [chap. VI. But /, zf{t)dt yt\t-Z) < \z m\ dt y\t\\t-z\' which is infinitesimal when the radius of 7 is infinitely great. Thus if we now suppose that the direction of integration along I is from — loo to + 100. Now if n be any positive integer and z be not equal to 0, — 1, — 2, etc., we have the identity 1 1 + ...+ z-t -^ • -8^(^+1) ' ^(^-i-l)(^ + 2) z{z-k-l)...{z + n){Z'-ty on substituting this in the second integral we have therefore Oi y(-^) = ao + ^ + (h z ziz-^-l) + ...+ ^'Wi where z{z-k-\) ,,.{z -^-n) ■^ 2in}iz (z + 1) ... IzJf n) (z - 1) ' /(t)dt t ^=2-^//(^>^^' Now the product can be written t(t+l)...(t + n ) z{z-{'l)...(z + n) t ** -n r and it diverges to zero or to infinity when z tends to 00 according as the real part of t—z is negative or positive, as can be seen by comparing it with the product 77] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 119 which has the value (n+ iy~'. But the real part of ^ — <« is, in the case under consideration, negative ; and so the product t(t+l)...(t + n) is infinitesimal when n is infinite. Since/(^) is finite along I, and / . *^' . is finite, we see that I z{Z'\-l)...(Z'^ n){z^t) is infinitesimal when n is infinite. We can therefore expand f(z) in the form >^<^>="«+ 7 + ^(^Ti) + ^(7+lHl+ 2) + - • the coefficients a being given by the above equation ; and this expansion is valid for all values of z whose real part is greater than the real part of z at any of the singular points of f(z), except for the points ^ = 0, -1. -2 Example 1. Obtain the same result by using the equalities «(«+l)(f+2)...(«+n) n\Jo ^ «/-««, , Example 2. Obtain the expansion where a„= r<(l-<)(2-<) — (^-l-0«fe> and discuss the region of its convergency. (Schl6milch.) Miscellaneous Examples. 1. Let er^Pn denote the nth derivate of e'^y so that P^,= l, P,= -2«, P,-4a»-2, etc Shew that if f(z) is an arbitrary function, then/(«) can be expanded in the form 1 r* where a-= | e-^Pn(x)f(x)dx. 2.4.6... 2nVir; — and find the region of convergence of this series. (Hermite.) 120 THE PROCESSES OF ANALYSIS. [CHAP. VI. 2. Obtain (from Darbouz's formula or otherwise) the expansion + ; find the remainder after n terms, and discuss the convergence of the series. 3. Shew that +( - D— ^•^•°-;-,j^""^^ J {/" (*+ A)+( - !)«/• (*)} ( - l)»h<'*' j\,{t)f*'{x+/U) dt, where y, (*)= -^^i-^ x»+i (\-x)n+i^{x-i (1 -a:)-*} rrfilj and shew that y^ (x) is the coefficient of n ! ^ in the expansion of {(1 — tx) (1 +<— tir)}'i in ascending powers of t. 4. By taking in Darboux's formula, shew that /(*+A) -/(x)= -a,h {/' (*+A)-l/' (*)| V 1 — r w tt' u' MISC. EXS.] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 121 5. Shew that 2! ~ ^^»^^;P<'-° ^r'(a)4-/-(.)} + 271 ! JV2m(0/<*'**U«+<(^-«)}<'^ -^ ♦.<')%-|r[^^'.(.t;)]...- 6. Prove that + (7«(«,-^)V'M^)+ where CJ^ is the coefficient of 2!* in the expansion of cot (^-5)1^ ascending powers of z, (Trinity College Examination.) 7. If ^1 and x^ are integers, and <^ (2) is a function which is regular for all values of z (finite or infinite) of which the real part lies between x^ and x^, shew (by integrating / 4>{z)dz £>2iriz _ 1 round a rectangle whose sides are parallel to the real and imaginary axes) that i*(*'i)+*(^i + l)+<^(a?i + 2)+...+<^(a?,-l)+i<^(^^ = ['*<f>{z)dz'k-l /'" ^(^tH-<y)-»(:Ci4-*y)-»(^,-ty)+»(j?,-ty) ^^ Henoe by applying the theorem where i?i, ^3, ... are the Bemoullian nimibers, shew that ^(l)+<^(2)+...+<^(n)-C+i^(n)+|**(«)c?^+^|^^-^^^^^ (where (7 is a constant not involving n) provided that the last series converges. 8. Obtain the expansion for one root of the equation x^2u+u\ and shew that it convei*ges so long as | :p | < 1. 122 THE PROCESSBS OF ANALYSIS. [CHAP. VI. 9. If /S^i, denote the sum of all combinations of the numbers 1«, 3«, 6«, ... (2n-l)« taken m together, shew that * ~8in»^^(2n+2)! l2«+3 "ifn+i) 2»+l* •" ^ »(»+« Sj 10. If the function /(<) is r^[ular in the interior of that one of the ovals whoee equation is | sin2|>'C (where C^ 1), which includes the origin, shew that /(<) can, for all points < within this oval, be expanded in the form - /P»)(0)+;S*2/(*-»(0)+...+^"-»/<'>(0) n')-m^^r — 2^1 '^•"' . g /'--''(0)+^^,./O'-')(0)+...+C/'(0) ^^...., where ^S^' is the sum of all combinations of the numbers 2«, 4«, 6«, ... (2n-2)* taken m together, and £f^^^^ denotes the sum of all combinations of the numbers 1«, 3«, 6«, ... (2n-l)«, taken m together. 11. Shew that the two series 2f^ 2^^ d 2g 2 / 2g Y 2 . 4 / 2^ y represent the same function in one part of the plane, and can be transformed into each other by Burmann's theorem. 12. If a function f(z) is periodic, of period 2fr, and is regular in the infinite strip of the plane, included between the two branches of the curve | sin «|kC7 (where 6'>1), shew that at all points in the strip it can be expanded in an infinite series of the form /($)= Aq-^- Aisin z-^ .,, + A^ain^ z-^ +coe «(i9i+5j sin «+...+5i»8in«-i «+...); and find the coefficients A and B. 13. If <f> and / be connected by the equation of which one root is a, shew that ^W ^ 1 ^^'•'^ ^2! <^'» <1>"{PF')' 1 1 ^ JL 31 <^'« 4>"' (*T' ( /'^T 1.1.2 T .••*.•, MISC. EXS.] THE EXPANSION OF FUNC?riON8 IN INFINITE SERIES. 123 where Fy /, F\ etc. denote F(a\ f{a\ -^- 14. If a function If (a, 6, x) be defined by the series IF (a, 6, ^)=a:+ 0-7-^+ which converges so long as 2! x\< 31 *•'+ h\: shew that dx^^""^ *' ^)-l+(a-fc)ir(a-6, 6, d?); and shew that if y « ir(a, 6, x)y then a;» IT (6, a, y). Examples of this function are ir(l, 0, ^)=^-l, If (0, 1, a:)=log(14-a;), (l + ;c)«-l 15. Prove that If (a, 1, 0?)^ 1 1 . ; (-ir^ g 00 n—O where Q. 4a^ 3a, 6a, 4a| 3ao (2n-2)aH-i : (n-l)ao na^ (n-l)a^., a. and obtain a similar expression for < 2 a,idf*> . (n-o ; (Jeiek.) (Mangeot.) 16. Shew that 2 CLfSf where /S^ is the sum of the rth powers of the roots of the equation 2a,.a?*'«*0. (QambrolL) 17. If /»(«) denote the »th derivate of /(«), and if /-»(«) denote that one of the nth integrals of f{z) which has an n-ple zero at < *0, shew that and obtain Taylor's series from this result, by putting g (s}b1. (Quichard.) 124 THE PBOCESSES OF ANALYSIS. [CHAP. VI. 18. Shew that, if ^ be not an integer, the series 2 (xH-m)*(arH-n)*' in which m and n receive in every poesible way unequal values, zero or integers lying between +/ and -/, vanishes when /increases indefinitely. (Cambridge Mathematical Tripos, Part I, 1895.) 19. Sum the infinite series 1 n— where the value n=0 is omitted, and p, q are positive integers to be increased without limit (Cambridge Mathematical Tripos, Part I, 1896.) 20. If F(x)^ei^'"^^'^^^, shew that n-l and that the function thus defined satisfies the relations F{x)F(l-x)^2amxw. Further, if ^(r)=:«+^ + ^,+ -J^og(l-«)rf(log4 shew that /'(ar)=c*'^2^*<^~'''*^^ when I !-«-«»*« I <1. (Trinity College Examination.) 21. Shew that [-(r][-(i^)l-G^J][-G-^J][-GA^)"] n I 2«(1-C08A')*«" * where (Lf^Jc Bin -^ — ir, n MISC. EXS.] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 125 " n and < j: < 2ir. (Mildner.) 22. If I ^|< 1 and a is not a positive int^er, shew that where (7 is a contour in the ^plane enclosing the points 0, x, (Lerch.) 2a If ^(«), <^,(«), ... are any polynomials in z, and if F(s) be any function, and if ^1 (')) ^a (^)) ••• ^ polynomials defined by the equations [* ^(4r) ,^, (*) *, (X) ... <^,_, (*) *"> )-»» W d«=^„(,), y a a— ar shewthat P ^W^ ,^iW, ^^aW *i W 4*% («) *s W 1 rF(x)<h{x)<h(x).,.4,^{x)^. J a t—X 24. A system of functions p^ (z\ p^ (z\ p^ (z), ... is defined by the equations where a» and b^^ are given functions of n, which for n^oo tend respectively to the limits Oand -1. Shew that the region of convergence of a series where e^y e^y ... are independent of «, is a Cassini's oval with the foci +!> - 1. Shew that every analytic function f(z\ which is regular in the interior of the oval, can for points in this region be expanded in a series where 126 THE PROCESSES OF ANALYSIS. [CHAP. VI. the integrals being taken round the boundary of the region, and the functions q^ {z) being defined by 26. If Pn (*) be the coefficient of — j in the expansion of in ascending powers of z, so that shew that (1) Pn {x) is a homogeneous polynomial of degree n inx and A, (2) 2-^-- (^^^)' (3) rPn{s)dx^O (n^l), (4) If y — o^Po (x)-^aiPi (^) H-OjPj (*) + .. ., where a^, o^, a,, ... are real constants, then the mean value of -j^ in the interval from x^ - A to x^m 4.A is o^. (L^ut^) 26. If P« (j?) be defined as in the preceding example, shew that />^-(-l)-2^^(^cos-j-2,^cos-^+3^cos-^ +...J, i'2»+i = (-ir2^^:^j(^sm-^-2^^jSm-^ + ^5;^jSm-^ + ...j. (AppelL) i CHAPTER Vll. Fourier Series. 78. Definition of Fourier series; nature of the region within which a Fourier series converges. Series of the type Oo + a, cos ^H-ag cos 2-^ + 08 cos 3£r + ... + 6i8iiiz + 6,sin 2^ + 63sm 3£r + ... , where Oo, Oi, a,, a,, fej, ftj, 6,, ... are independeat of z, are of great import- ance in many analytical investigations. They are called Fourier Series, We have already seen that the region within which a series of ascending powers of z converges is always a circle \ and the region within which a series of ascending and descending powers of z converges is the ring-shaped space between two circles \ we are therefore led by analogy to expect that series of the Fourier type will likewise converge within a region of some definite character. To investigate this question, write 6**= f. The series becomes This is a Laurent series in f ; it will therefore be convergent, if at all, within a ring-shaped space bounded by two circles in the ^-plane ; that is, it will be convergent for values of f satisfying an inequality of the type • a<|?|<fe, where a and h are positive constants. Now let £r = a: -h ty ; then 128 THE PROCESSES OF ANALYSIS. [CHAP. VU. aod therefore the inequality becomes log a< — y< log 6. This inequality defines a belt of the ^r-plane, bounded by the two lines y = — log a and y = — log h ; hence the region of convergence of a Foui-ier series is a belt of the z-plane, bounded by two lines parallel to the real axis. It may however happen that the Laurent series in f is divergent for all values of f, in which case the Fourier series is divergent for all values of z ; or, (and this is the most important case for our purpose,) it may happen that a » 6, so that the region of convergence of the Laurent series narrows down to the circumference of a single circle in the ^-plane ; in this case the region of convergence of the Fourier series narrows down to a single line parallel to the real axis in the plane of the variable z. If now the coefficients Oq, Oi, a,, ... 6], &t, ... are all real, considerations of symmetry shew that if the Fourier series is divergent for a value z^a + tb, it will also be divergent for the value -zr =s a — i6 ; so if in this case the region of convergence narrows down to a line, that line can only be the real axis in the ^-plane. Hence a Fourier series with real coefficients may converge only for real values of z, and diverge for all complex valves of z. An example of this class of expansions is afforded by the series sin-^ — 2®^^2^+3^^3^"" 4 s^^ 4^+ ... . Writing this in the form we see that it diverges when z is not purely real ; when z is purely real and not an odd multiple of ir, the sum of the series is ilog(l+6'0-^.log(H-6-^X or 21'^^^**' or i^z-Vkir, where k is some integer, as yet undetermined. Now when z = the sum of the series is seen directly to be 0; when z^-ci. the sum of the series is tan~^ 1, or 7- ; when ^ = — — the sum is 2 4 2 78] FOURIBR SERIES. 129 — tan""* 1, or — 7 . In this way we see that when z lies between — ir and + ir, the integer k is zero. But A? is no longer zero when z is greater than ir ; for each term of the series is clearly unaffected if ^ + 27r be written for z : hence the sum of the series must be the same for -^ + 27r as for ir ; and hence when wkzk Stt, the sum of the series is ^z-^tt; so that when z lies between ir and Stt, the integer Ar w — 1. Proceeding in this way, we see that the sum of the Fourier series is ^z + kir, where k is an integer chosen so as to make ^z + kir lie between — ^ and + ^ . This is important as shewing that the sum of a Fourier series is not necessarily a continuous analytic function. It is clear however that the sum of a Fourier series can have discontinuities only in the case in which the region of convergence narrows down to the real axis ; in the other case when the region of convergence is a belt of finite and infinite breadth, the Laurent series in ^ represents an analytic function, and therefore the Fourier series in z does also. Example. Shew that the series cos «- 52 cos 2« + 5j COS3-5- ... n^ 1 converges only for real values of 2, and that when -ir<«<-firit8 sum i^ ts - 7 **• For when z is real, the series is absolutely and uDiformly convergent, as is seen by com- paring it with the series 1 + 02 + 02 "*"••• • When z \a complex, we have (putting z=x+iy) -ijC0srw=^ {e*(»«+"*»')-|-e<(-»«-»«*')} ; now either — , or ^ s ^ infinite for n=ac, so the terms of the series are ultimately infinitely great and the series diverges. To find the sum when z is real, it has been shewn that when -7r<z<ir we have ^z=:8in z- ^ sin 2i+i sin 3;? ... . This series is imiformly convergent in the interval (though not at its extremes - ir and fr) and so can be integrated. Thus c-J«*=cos;8-5jCos2a+^cos3«-,.., where c is a constant. W. A. 9 130 THE PROCESSES OF ANALYSIS. [CHAP. VII. To find c put «»0, which gives whence the result. ,1.1 _^ ^" 2«"*"3«"*""12' 79. Valties of the coefficients in terms of the sum of a Fourier series, when the series converges at all points in a belt of finite breadth in the z-plane. The connexion between the coefficients a^, Oi, a,, ... , 6i, 69, ... of a Fourier series, and the sum of the series, can be easily found in the case in which the series converges in a belt of finite breadth in the ^-plane. For in this case, as we have seen, the sum of the series is an analytic function of z. Let it be denoted hyf{e), so that f(t) ss a^ + a^cos z + a^coa 2z + ... +6, sin 5-^69 sin a? + — Writing f = e^, the series becomes and by Laurent's theorem the coefficients in this expansion are given by the equations 2^^-^^j /(^)?-»d?. where C is any circle in the (-plane, surrounding the origin and contained within the ring-shaped region in which the expanded function is regular. Now if the quantities Or and br are all real, we see as ^before by symmetry that the real axis must be contained in the region of convergence in the jr-plane, and therefore the circle of radius imity must be contained in the region of convergence in the (-plane, since this circle corresponds to the real axis in the jff-plane. We can therefore take (7 to be a circle of radius unity, with the point (b as centre. Now writing f « e^ in the integrals, we have and so (ar + ibr)^j^y(z)(f^dz, 1 r*» ttr = - / f(z) COS rzdz (r > 0), 1 ri» . 6r « — I f{z) sin rzdZf 79, 80] FOURIER SERIES. 131 and ^"""^ZttJ f^^^^' These equations give the vahies of the coeflScients Oo, Oj, a,, ... , bi, ftj, ... , of the Fourier series, in terms of the sum f(z) of the series, in the case in which the series converges over a belt of finite breadth in the 2:-plane. We shall see in the next article that the same formulae hold good in the more extended case, in which the series converges only for real values of z. sme JBaxunpU, Shew that the function , f^ can, when k<l, be expanded in a Fourier seriee of sinee of multiples of «, valid for all points z situated in a belt, of width -2 logir, parallel to the real axis in the ;B-plane. For we have sin a = Li_i 1 .1 00 I and this can be expanded in the form 2 i, /;* (6***-e-**»), provided k le**! and ;t |e-*»| are less than unity. This can only happen when their product X^ is less than 1, Le. when -l<it<l. When this condition is satisfied, on putting z^x-^iy, it is clear that we must have e*«-i'|<T and>it, Le. we must have-y lying between logfi) and log it, Le. z must be within a belt of width -21ogiI;, parallel to the real. axis. When these conditions are satisfied the expansion is valid, and so 8in« * . , . 2 A!* *smn«. l-2irco8;j+it* Hal 80. Fourier's Theorem. ft We have already said that the most interesting cases of Fourier's series are those to which the investigation of the last article cannot be applied, on account of the fact that the series converges only for real values of z. It is therefore necessary to undertake another investigation, in which the assumptions of the last article are no longer made. The result to which we shall be led is known as Fourier's theorem, and may be stated thus : Iff (z) be a quantity which depends on a variahle z, and which is finite and has only a limited number o/maaima and n^inima and of finite discontinuities in the interval <z< 2ir, then the sum of the series CO ^0+ S (amOosmz + bmsiamz), mai 9—2 132 THE PROOBSSES OF ANALTSia [CHAP. VII. where 1 f *» dm = - / /(t) COS mtdt, 1 r*» 6m= - / /(O sin TYVtdt, repre^ento y (2:), at every point in the interval < ^ < 2^ for which f(z) is continuous; and at every point in the interval 0<z<27r for which f{z) is discontinuov^Sy the sum of the series is the arithmetic mean of the two values of f(z) at the discontinuity. The diacussioQ of Fourier's theorem given below is a modification of what is known as Cauch^i second proofs which was originally published in 1827 in the second volume of his Exercices de Math^moOiqueif and is reprinted in his Collected Works, Second Series, YoL vn., p. 393). This proof, (which in its original form was in some respects imperfect,) seems to have been little used by the mathematicians of the nineteenth centiuy, who in the discussion of FourieHs theorem almost universally followed the exposition of Dirichlet (which is also reproduced later in this chapter) ; the importance of Cauchy's proof was shewn by A. Hamack in 1888. It may be observed that the restrictions placed on /(«) — as to its having only a limited number of maxima and minima, etc. — are suficieni but not necessary for the validity of the expansion. To establish the theorem, we write the first 2ik + 1 terms of the expansion in the form ~l f(t)dt'^^ 2 f{t)coam(Z''t)dt, or or . Uk-h Fjk, where Um = "*2* ^ f V* <*- V(0 dt, We shall now investigate the behaviour of the quantity J7fc when k, though finite, is a large number. Let <t> (^) denote the quantity 80] FOURIER SERIES. 133 Then <f> (^ clearly has a definite value corresponding to every value of ^, except the exceptional values ^=0, ±i, ± 2t, for which c*^= 1 ; moreover, it is easily seen that the quantity tends to a definite limit when B^ tends to zero, independently of the way in which Sf tends to zero (still excepting the points 0, ±i, ±2i...); hence <f>{^) is an analytic fiiDctioD of f, having poles at the points 0, ± t, ± 2i ... ; and the series U^ is clearly the sum of the residues of ^ (^) at those of its poles which are contained within a circle C> in the ^-plane, whose centre is at the origin and whose radius is (* + 2) • Hence Write Thus Now we can write .i-»-» +»-» j^.i-i »jr+»-i ir , , _ ^*==2Uf +f +f +r +r u*(^^^ «/i + /, + /3 + /4 + /b say. At points in the range of integration of /, and I^, the real part of ^is positive and at least of order k'^; and so for these integrals we have In this expression, as k tends to infinity, ^^-^^^ tends to the limit unity, and {'e^(»-<-*r) tends to the limit zero : thus ^<f>(^ tends to the limit zero, since the range of integration from to ^ is finite ; and hence as /i and I^ are the integrals of ^(f> ((^ taken over finite ranges, we see that /i and I^ tend to zero as k tends to infinity. Considering next the integrals /, and /«, we observe that the quantity ^^_^ is never infinite when < ^ < 2: < 2'jr, and so f^ ((^ is never infinite ; 134 THE PROCESSES OF ANALYSIS. [CHAP. VIL and thus, since /{ and /« are integrals of ^(f> (^ over ranges which become infinitesimal as k tends to infinity, it follows that /, and /« tend to zero as k tends to infinity. Consider next the integral /j, or where f* (?) « ^^jjeii^t,f(t) dt In the range of integration of /«, the real part of ? is negative and at least of order W. The feu^tor -r-? — r- therefore tends to the value — 1 as fc 6*^— 1 tends to infinity ; denote it by — (1 + aj^), where o» tends to zero as k tends to infinity. Now r^(^^*^/(t) dt^J, + /„ ^0 where /,«[ ^'^ l:^^'^fit)dt, and t7,= r ^ K^^-^f{t)dL '"log* Considering first /i, we see that within its range of integration the quantity ^{z-^t) has its real part always negative and at least of order '. — T, which tends to infinity with k\ hence the quantity ^ef^^^^ tends to zero as k tends to infinity ; and therefore as the range of integration in t/^ is finite, we see that Ji tends to zero as k tends to infinity. Consider next J^. Writing t; ■» ? (-«r — e), we have log* and writing e^^Wy this becomes /•log* / ^\ J.^j^''f[z-'2^)d.. Now as k tends to infinity, 6*^* and — ^ tend to the limit zero. Let /(^— 0) denote /(^) if ir is a point at which the function f{z) is a continuous 80] FOURIER SERIES. 135 function, and at those points at which f(z) is a discontinuous function let f(s — 0) denote that one of the two values oif{z) which is continuous with the value of/ for values smaller than z. Then since there cannot be another discontinuity within an infinitesimal distance of z, we can write /(-^)'/(*-0) + ,. r where 17 tends to zero aa k tends to infinity ; and so f -L. [e fe t7,=/(^ — 0)1 dw+l lydiw; jf - log* = -/(^ - 0) + e'^^'fiz - 0) + j ^ vdw, or J,=:-/(^-0) + €, where e tends to zero as k tends to infinity. Thus l;<l>(0 (l + «*){/i-/(^-0) + £}, where a», Ji, and e, each tend to zero as k tends to infinity. We can write this ^<f> ((^ ^/(z — 0) + T, where t tends to zero as k tends to infinity ; and this is true throughout the range of integration of the integral is- Thus or /, = 5 /(z — 0) + a, where <r tends to zero as k tends to infinity. Hence finally I7» = 5/(-^-0) + <r+/, + /. + /4 + A, where <r, /i, /,, /«, I^ each tend to zero as k tends to infinity ; which can be written where Uj^ tends to zero as k tends to infinity. Similarly we can shew that where v^ tends to zero as k tends to infinity, and where /{$ + 0) denotes f(z) 136 THE PROCESSES OF ANALYSIS. [CHAP. VH. if 2^ is a value for which the function/(^) is continuous, and denotes the value oif{z) for values slightly greater than 2: if 5 is a value for which the function f{z) is discontinuous. Hence the sum of the first (2A; + 1) terms of the Fourier series is \f{^ - 0) 4- ^/(^ + 0) + w* + V*, where uj^ and v* tend to zero as k tends to infinity ; the sum to infinity of the series is therefore ^{/(^-0)+/(^ + 0)]. which establishes Fourier's theorem. It must be observed that the sum of the series coincides with f{z) only far values of z between and 27r ; outside these limits the sum S (z) of the series can be found from the circumstance that S{z-\'2nir) = S(zX (a result which is obvious, since all the terms are periodic) ; while f{z) may of course have any values whatever when z is not included between the limits and 2^, Example, Take a function /(a) such that IT f{z)wm- from «««0 to «=ir, and /(«) = -J from;? « IT to «=2ir. The corresponding Fourier series is Oo + SOm OOS97U + 26m siu mz, where O'm^^- \ f(t)co3mtdt, n J 'm 1 r«» -i /(Osi irjo sin mtdt, These integrals give 1 f» 1 f^ 00=0, Otn'^i I cosmtcfe-- / cosmtdt^'Oj 6«=T / sinm^(ft-7 I nin mtdt=-r- (I -cos mw). 4;o 4j» 4m'' Therefore 6»,=0 if wi ia even, and 6„»= — if m is odd ; and so we liave J, . sinz sindi? . sinSz . which is the required Fourier expansion. 81] FOURIER SERIES. 137 This series can be summed by elementary methods in the following manner. We have . ^sinat^sin52 1 ( ^^^ ^ \ 1 / ^^«-w^ \ 4i*^(l-e<')(l+fl-^) 41^"^ 4+ 2 » where r is an undetermined integer. It is clear from the above that r actually has the value zero when < 2 < ir, and unity when w<z<2w. 81. The representation of a function by Fourier series for ranges other than to 2^. Suppose now that the range of values of z, for which it is required to represent a function f(z) by a Fourier series, is not the range from to 2^, but from a to b, where a and b are any given real numbers. To extend Fourier's result to this case, we take a new variable z defined by the equation and write 6 — a / •^(«+-^^)'^<'^- Then F{/) is a function whose value is given for all values of its argument y between and 2^. Therefore by the previous result we have ^(i^ = A fV(Od^+- 2 f*'cosm(/-0^(0^', or writing we have 6 — a , /«-^J>)*-E^,.!j> ^™t5-/«^ This last result may be regarded as the general form of Fourier's theorem. • Example. To express the function — ^ — ^^ as a Fourier series, valid when -ir<z<ir. Here a™ — ir, 6«»r. 138 THE PROCESSES OF ANALYSIS. [CHAP. VII. The formula therefore becomes /W-i r /(0^+; 2 f' coBn{z-t)f(t)dt. ^*r y -r w nail J -» Since in this case / (O— -/( - 0» ^^ reduces to /(«)=- Z sin Yu / — --sinn^(i< •■ 2 sm nj I ^ . . ^i _^ ~ dt ii.iirt («**»- e~"^) I m+tn m-in J • (-l)»sin nz / 1 _ 1 \ HBi »r(m'+n«) ^ which is the required expansion. 82. The Sine and Cosine Series, We proceed to derive two particular cases of Fourier's theorem which are of frequent occurrence. Suppose that a function f(z) is given for a range to 2 of values of the variable z, and that we require a series which shall represent f(z) for these values of z^ and which shall have the value /(— z) for values of z between and — L To obtain a series of this character, we write in the preceding result a » - Z, h^ly /{—z) ^f(z). Thus we have /(*) - ^ijjit) dt + ] J J'^cos^'^-^l^- *>/ (*) dt, / w - i/Vw '^ + T A «» ^£ «» ^y (0 dt. which is called the Cosine Series. If on the other hand we require a series which shall represent /(^) for values of z between and I, and shall have the sum — /(— z) for values of z between and — I, we write in the general result a = — Z, 6 = hfi" z) =» — /W> and thus obtain f(z) « 7 2 sm -J- I sm j-f(t) at, ^ m-l f' Jo ^ which is called the Sine Series, 82] FOURIER SERIES. 139 If ^ z EsxanpU 1. Expand -—^ sin « in a cosine series, valid when 0<f <fr. When < f < IT, we have by the formula just obtained -sinf-s^- I (w-t)smtdt+- 2 cosmi / {w''t)BmtooenUcU «r y IT iM>i J 2 •i[o"^'"~']"i/o'^^^ 1 • + 5— 2 OOSf/tf |r (fr-08in(m+l)*(ft- j (ir-t)mn{m-l)tcu\ 1.1 ^1 1 r ^ ^ 1 ■■S + l<5<»'+S- 2 cos HI* >-= y 11 * cosms 5 + 70081- 2 2^4"^' ^,(m-l)(m+l)- The required series is therefore It will be observed that it is only for values of « between and w that the sum of this series is proved to be —^ sins; thus for instance when « has a value between and — ir, the sum of the series is not ^^^— sin^ but - ^5— sin g; when « has a value between ir 2 2 and 2ir, the sum of the series happens to be again — ^— sin s, but this must be regarded as a mere coincidence arising from the special function considered, and not from the general theorem. Example 2. To expand ^-g— - ^ * «^« series, valid when 0<«<ir. We have — ^ — - ■■ — 2 sm 9IU I — ^-5 — •' am nUat o n fli*i Jo o ■■ 2 smini J -^— ; — ^ BinnUdt. Mai 'io-4 r«(ir-20ffln««n ^ 1 /■» . ,j. Therefore 2m« vziw'-t) . sinSf . sin5f . 140 THE PB0CES8ES OF ANALYSIS. [CHAP. VTI. Here again the sum of the series is -^ — ^ only when z lies between and n. Thus when s lies between «• and 2»r, the sum of the series is — ~ — . The sum of the o series for values of z beyond the limits and ir can be found at once from the equations S (t)=» - S {" z) and S{z+2ir)^S(z)y where S(z) denotes the sum of the series. Example 3. Prove that, when < « < ir, yr (tr ~ 2z) ( w' + 2nz - 2^) cos3« cosSa 96 -cosa+-g^ +__+.... For when < « < «• we have — ^ ^-i-jr^ i«- 2 cosm« I -> ^-^ "^cosm^cb 96 fTfwsi Jo 96 (integrating by parts) = 2 cosm« I sinm^cfe ■1=1 J 4^ (integrating by parts) = 2 cos me J ~-n- cos mtcft m=i Jo 4m' » /■» 1 (integrating by parts) = 2 cosm;; ) r— .sinmtcfo mat y *^ « i-(-i)- * — s — 2 — oosm* m-i 2m* . cos Zz . cos 5e eoe f +— ^ + -^j~ + ... . Excm/ple 4. Shew that for values of z between and ir, «** can be expanded in the cosine series and draw graphs of the function «** and of the simi of the series. Example 5. Shew that for values of z between and ir, the function — ^^-^ — - can be expanded in the cosine series . cos Sz . cos 52 and draw graphs of the function -^-^ — < and of the sum of the series. 83. A Itemative proof of Fourier's theorem. Another proof of Fourier's theorem, based on an entirely different set of ideas, is due to Dirichlet*. ♦ CoUecUd W&rke, Vol. i. pp. 188—160. 83] FOURIER SERIEa 141 CoDsider first the sum of a limited number of terms of the series 00 where a^^- lf{t)coafntcU (m«l, 2, 3, ...), IT J 6^=i [^/(Osinm^cft (m-1, 2, 3, ...), V J and where z is supposed to be a real variable. Since V J we have the sum to (2m +1) terms of the series expressed by the formula ^m-- /{i+coe(^-«)+cos2 (<-«)+. ..+oosm(<-«)}/(Oflfe ^ /'2»sin(2m+i; h sm — «r j , sin ^ J \ -^ I 2 =1 p5£(??^V(^+2^dd fT J sm ^ -^ ^ ' 1 p- Bin(2».+l)0 IT jo sm^ -^ ^ ' We have therefore to investigate the value to which integrals of this class tend as m tends to infinity. Consider in general the value to which /./*«£*!^(.)& y sm « tends when h^ supposed to be an odd integer, increases without limit. First suppose 0<A<5 , and suppose that, for values of z within this range, ^ {z) is continuous and positive, and that <^ {£) continually decreases as z increases. Let -T- be the greatest multiple of t in A, so -r- < A < r+1 t . 142 THE PROCESSES OF ANALYSIS. [CHAP. Vn. Then » fw (n+l)ir f» Now write k nw k + ...+ / •* (r-D- k k Binkz ainz n» f * sin \kz 'it\fl» so U»' ir f * siniy - /nir , \ , , nv I . /n7 \ »(,T-^y;^' where y=.--p. The integrand in this last integral is clearly positive throughout the range of integration, and Uj^ is therefore positive. Moreover, under the suppositions already stated, the quantity decreases as n increases, and it therefore follows that u^ decreases as n increases. Also the well-known theorem of Mean Value shews that tin can be represented in the form where ir fk ainh/ . and Pn-^f^+^K S being some quantity between and r. Clearly y^ is positive, and decreases as n increases. Now we can write Jrw smz ^^ *^ k where J'=Wo-^i+t«|-Ws+...+(-iy"*Mr-i. ■ Since u^ is always positive, and decreases as n increases, we have r-1 where m is any number less than —^ . This gives < "OPO- («'l-^l)Pt- ("8- »'4)P4- — -("jm-l - O Plm -'l(Pl -P8)-»'8 (Ps-Pi)- — - "jm-lCPlm-i-Ptm). 83] FOURIER SERIES. 143 As p^ decreases with increase of n, the terms in the last line are negative, and can be removed without affecting the inequality. Thus •^< ^oPto - ("l - "l) Ps - el's - •'4) P4 - • • • - (•'jm - 1 - I'm.) Ptm <''0P0-(»'l-»'8)Plin-(»'8-^)Ptm-...-(''lm-l-Vjm)Pfm -(»'l-''8)(Pt-Plm)-(»'3-l'4)(p4-Plm)-...-(»'jm-8-«'jm-8)(p2m-2-p2m). The terms in the last line are again negative and can be removed. Thus •^< "0 (PO - Pjm) + (''O - "1 + ''8 - • . • + ''Sm) Ptm. We also have clearly J'>tio-tii+w,- ... - 1^.1, or •^>«'oPo-''iPi+''8P2- — -^jm+iPiw+i, which in the same way gives •^>Pfm (''0- 'l + ^S" ••• ~ "Jm+i)- Thus J" is intermediate in value between the quantities and Pim(''o-''i+*'2- — "-''8m+i). Now let k become infinitely great, and let the quantity m likewise become infinitely great, but in such a way that t tends to the limit zero. Then the quantities pQ and pi^i tend to the limit ^ (0) ; and the quantity (am-f 1) V f * sinitv J or / -: — '- dv Jo smy ^ mm+Dir sint . or 1 ^ cU Jo 7 • * •^ it; sin T k can, since k is infinitely large compared with m, be replaced by / (2in+l)»8in t ,, * and this, when m becomes infinitely great, tends to the limit ^ . We see therefore that J is intermediate in value between two quantities, each of which tends to the same limit, namely ^ ^ (0). J therefore tends to the limit q<I>(0); and therefore /, which difiers from J only by a vanishing integral, likewise tends to the limit Q<l>(0) as k becomes infinitely great. This result may be called Dirichle^i lenvma. To complete the lemma, however, it will be necessary to shew that it is still true when a number of the restrictions imposed on ^ {z) are removed. (1) It was assumed that if> (z) was positive and steadily decreasing throughout the range. (a) Suppose that ^ (z) is constant This does not invalidate any of the preceding proof, so the theorem still holds if ^ (1) is constant 144 THE PBOGBSSBS OF ANALT8I& [CHAP. Yll. O) Suppose that ^ (z) is negative, or partly positive and partly n^^ative, but still steadily decreasing; then choose a constant o so that c+^ (e) is positive through the range; then the theorem applies both to o and to o+^ («) and therefore on subtraction to (i) alone. (y) Suppose that <f> (z) increases steadily throughout the range. Then the theorem is true for { — <^ (z)} and therefore for <^ (z). Therefore the theorem is still true if <^ (z) is ftmte^ continuous, and steadily increases or decreases throughout the range. (2) Instead of taking the integral between and A, take it between g and A, where 0<g<h^-^. We assume that the value of (t>{z) is only known for values of z from ^ to A. Take a new function ^ («), defined as being equal to <^ (g), a constant, for values of z from to ^, said equal to (z) for values of z from gU> L Then the theorem holds for 0^ («). Also ^* /la* <^i(«)<&-|*i(0)=|<^(!7), and Therefore by subtraction. (3) Now assume there are a limited number n of maxima and minima within the range to A. Let them be at the values Oi, a^, ... 0^, of s. Then On applying the theorem to each of these integrals in succession, it is clear that the theorem holds for the whole integral. Therefore the theorem is still true if <f) {z) is finite, continuous, and has not more than a limited number of maxima and minima within the range. It must be noted that these conditions still exclude such functions as e.g. (^-c) sin — z^ c where 0<c<A. IT (4) We shall now no longer restrict A to be less than 5- . Take < A ^ ir. Then(a)let^<A<tr. WriteA=tr-A', where < A' < | . Then Lamit /=* / -: <p («) dz + I —. — <f> (z) dz, j^moo J sm z J ^ SID z 83] FOURIER SERIES. 145 Writing z^n-Cin the latter integral, we have ir » Limit /« P2iji^*Wcir+ P«^!^*;0(.-f)eif. Since <l>{fr-() satisfies the conditions stated, we see that when A' > the second integral is zero. ^ Therefore Limit /- ^ <^ (0). O) Let As TT. Then all the above reasoning applies, except that now A' 1*0, so Limit/«|^(0)+f<^(ir), which, in order to guard against uncertainty in the case in which the function is discontinuous at and ir, is often written where ff is a vanishing positive quantity. (5) Next, suppose that the function <f) (z) within the range has a finite number of dis- continuities, in the form of abrupt but finite changes of value. Divide the range into various portions, so that each of them ends at one discontinuity and begins at the next, and divide each of these into others each beginning and ending at a point of stationary value. The above theorems apply to each of the portions, and therefore each integral is zero except the first, which is equal to o^(<)) ^^d possibly the last, which when h^n has the value o^Ctt-c). (6) Finally, consider a function <t>{z) which becomes infinite for z^se, but in such a way that the value of / ^ (z) dz tends to a definite limit as z approaches c from either lower or greater values. Then where c is a small positive quantity. In the second integral, a quantity ( can be chosen intermediate between c and c- c, such sin JkC f^ that the integral is equal to -^— ^ / ^ («) c& ; on taking e small this vanishes ; and sin C y c - • similarly the third integral is zero. On making k infinitely large, the fourth integral tends to zero. Therefore the theorem holds in this case also. (7) Thus we have, summarising the results obtained, the theorem that the Umit when jk tends to infinity of I ^^ <l>{z)dzi8^<l){€)ifO<h<Viandis Jo sm z 2 W. A. 10 146 THE PROCESSES OF ANALYSIS. [CHAP. Vll. if h^it ; xohere € is a vanvUhing positive quantity ; provided thai <ft{z) is every where finite, and has only a limited number of finite diecontinuitiee and maxima and minima between the values and h of the variable z ; and this is still true if <f> {t) has a limited number of singularities of specified type, namely such that I <f>{z)dz is finite. This result may be called Dirichlefs lemma, the conditions just stated being referred to as Dirichlefs conditions. We can now return to the expansion which was found to represent the sum of the first (2m + 1) terms of the Fourier series. We had iS'w=/i+/„ X. r 1 /*'''«sin(2wi + l)tf ., , ^A^ jA where /.--j^ ^__i_y(,+2tf)rf^, 1 r*«n(2^) ^ n Jo am tf -^ ^ ^ If 0<;r <29r, and/(j;) satisfies Dirichlet's conditions, we have hj Dirichlet's lemma Limit /i— i/(«+€), and Limit /a=i/(2-€), and so Limit 5„-i {/(^+,) +/(«_,)}. It z^O, we have Limit /j = i {/(f) +/(27r - c)}, Limit 7,-0, and so Limit &,=:i {/(c) +/(27r - f )}. If «=2ir, we have Limit /i=0, Limit /,-*{/(«) +/(2ir-€)}, and so Limit 5„=i{/(«)+/(2,r-€)}. Thus we finally arrive at Fourier's theorem, namely that the sum to infinity of the series cio+ S (a^cos9iu+6mSinnu) is f{z) at points z for which f is continuous, and is the arithmetic mean of the two values off{z) at points z for which f is discontinuous : it being assumed that f{z) satisfies Diriohlet's conditions. Example, Prove that in the limit when n becomes infinitely great /'•8in(2n + l)0 .^,. , ... a being a real positive constant. (Cambridge Mathematical Tripos, Part IL, 1894.) 84] FOURIER SERIES. 147 84 Nature of the convergence of a Fourier series. The proofs of Fourier's theorem which have been given establish the result only for the case in which the sequence of the terms in the series 2 (dm cos mz + bm sin mz) is that in which m takes the orderly succession of values 1, 2, 3, 4, ... . The question now arises whether the order of succession of the terms can be deranged without affecting the value of the sum of the series ; in other words, we have proved that the expansion of a function by Fourier's theorem is a convergent series : we want to find whether it is absolutely convergent, or only semi-convergent. The question has also to be considered whether the series is uniformly convergent or non-unifomdy convergent in the neighbour- hood of a given value of z. We shall first shew by considering special cases that there is no geneml answer to these questions. Consider the series sin i^— -sin 2^ + ssin3-«f— ... , which represents - z when Okzktt, and ^z^tr when ttkz <2Tr; this series is semi-convergent for all real values of z, since sin n.^ is finite for all values of n when z is real, and so the modulus of the general term bears a finite ratio to the general term of the divergent series In this series, therefore, the value of the sum will be modified if the order of succession of the terms is changed. Moreover, we can shew that the series is non-uniformly convergent at its discontinuity tt. For the sum of the first n terms is sin 2z . (- 1)***"* sin nz smz 5 I-...H , 2 n or / (cos e - cos 2e + ...+(-!)'*"* cos 7rf)cie, Jo [* fl (- !)**-» co8 (n + l)^-hcosnel , Jo [2 "^ ~~2 nr^^^t 'J ^^- The term I ^ dz represents the sum of the whole series ; so the remainder Jo 2 after n terms, when — tt < z < tt, is or cosf»+ gj t iin«(-ir'| ^ ^ dt. 2C08^ - 1)"-' io 10—2 148 THE PROCESSES OP ANALYSia [CHAP. VII. Writing <r = 7r — i;, ^■btt — w, this can be written i2n=-| sin (n+ gj u dtL 28in^ Write (n + i j u « I'. The equation becomes Sin V dv. However great n may be taken, if 17 be taken so small that (^ + 5)^ is infinitesimal, this int^;ral tends to — I or — -5 , and so is not infini- tesimal It follows that the series is non-uniformly convergent in the vicinity of -^ — TT. Consider next the series 1 0.1 COS<r + — C08 3ir + r:C0s5<r + .,. , o' o' which represents ^^ q ^ when Okzktt, and — — ^ when 7r<z<27r, This series is absolutely convergent for all real values of 5, since the moduli of its terms are less than the corresponding terms of the convergent series 1 1 1 + g, + gj + . . . . In this series therefore the order of succession of the terms can be changed in any way, without altering the value of the sum of the series ; and since the comparison series is independent of z, the series is also tmi/orrnlj/ convergent for all real values of z. Returning now to the general Fourier series, we can discover the nature of the convergence by a. consideration of the coeflBcients in the series, which can be made in the following way. We have shewn that if ao then 1 f *» a,» = - 1 f(t) cos mtdt 84] FOURIER SERIES. 149 Suppose that (as in most of the examples we have discussed) the range 0<z<2Tr can be divided into other ranges, say 0<z<ki, ki<z<k^, ,,,y kn<z< 27r, which are such that in each of these smaller ranges f(z) is an analytic function of j?,, regular iu the range. (f(z) will not necessarily be the same analytic function in the different ranges.) Thus it f(z) has the value J? for < J? < TT, and has the value --z for ir <z< 27r, we should have w « 1 and ki = tt. Then 1 r *» ,1 /**« 1 f** am=- I /(t)coBmtdt + - f(t)coamtdt-\' ... + - I f(t)co8mt(JU. Each of these integrals can then be integrated by parts ; we thus obtain [*» 1 J.... sin mf\ , r*« 1 ^ ... sin vitl . 1 r*« 1 f*« / f'(t)smmtdt I f'(t)sinmtdt^ ... , TrmJo ''rmJkf ^ or where il = ~ [sin mJe, {/(k, - 0) -/(A, + 0)} + sin mk, {/(k, - 0) --/(k, + 0)} + ...], and where bm is the coefficient of sin mz in the Fourier expansion of/' (z) — an expansion which will exist, since/' (z) is a function of the same character as f(z)y though the terms of this expansion will not always be the derivates of the corresponding terms of the Fourier series {ot/{z). Similarly 6 =:? + ^' m m where -B«--[-/(+0) + cosmA;j{/(*,-0)-/(A^ + 0)} + co8mti{/(ifc,-0) TT -/(A, + 0)} + ...+/ (27r-0)], and where a^' is the coefficient of cos mz in the Fourier expansion o{f'(z). In the same way we have m m where il' = ^ [sin 7n*i (/' (jfei - 0) VX*i + 0)} + sin TnJfc, {/' (ifc,- 0) V' (*«+ 0)} 150 THE PROCESSES OP ANALYSIS. [CHAP. VIL and , , B^ .dm where dm" and 6m ' being the coeflficients of cos mz and sin mz respectively in the Fourier expansion of/'' {z). Thus A B! a^ // , B A' bm" 6w =» - + — 1 r . The conditions for the absolute convergence of the Fourier expansion of f{z) are therefore expressed by the equations il-0, 5 = 0; for if these equations are satisfied, we have a„ = .^?l±^'and6, ^'"^ m* *" m* and the terms of the Fourier series are comparable with those of the con- vergent series 1+1+1+1+ Now in order that we may have ^ = 0, B = 0, for all values of m we must have /(*.-0)=/(A, + 0), /(Ar, - 0) =/(*, + 0). That is to say, if a Fourier series is absolutely convergent for aM real values of z, the fwnction represented by the series has no discontinuities, a/nd has the same value at z^O as at z^ 27r. If these conditions are satisfied the Fourier series is not only absolvlely, but is also vmformly convergent For its coefficients a^ and 6m are in this case of the order — ; , and so the series of constants ^ol + jchl + 16, ! + |a,| + |6, 1 +... 85] FOURIER SERIES. 151 converges ; but the moduli of the terms of the Fourier series are less than the corresponding terms of this series, and consequently the Fourier series is uniformly convergent for all real values of z. Example 1. Shew that in general, when the Fourier series converges only for real values of t^ the quantities a^ and h^^ can be expanded in infinite series of the form of which the terms m w' m' tn* A B .B A' 5 and - + — , found above are the initial terms ; but that when the Fourier series converges within a belt of finite breadth in the «-plane, all the coefficients c^) c,, C3, ... vanish, and this expansion becomes illusory. Example 2. Let /(«) be a function of «, which is regular for all real values of z between ««=0 and «=7r, and which is zero at «=0 and z^rr. Prove that if /(«) is expanded in a sine series, valid between z=0 and z^fry the series will be absolutely and uniformly convergent for all real values of z. Example 3. f{z) is a function of z which is regular for all real values of z between and TT. Prove that if it is expanded in a cosine series, valid between zs=:0 and z^n, the series will be absolutely and uniformly convergent for all real values of z, 86. Determination of points of discontinuity. The expressions for dm, and 6^ which haye been found in the last paragraph can be applied to determine the points at which the sum of a given Fourier series is discontinuous. This can best be shewn by an example. Example, Let it be required to determine the places at which the sum of the series sin £+} sin 32+i sin 5^+ .•• is discontinuous. For this series we have 1 «m-0, , 1 - COS mit *»'- 2m" Comparing this with the formula found in the last paragraph, we have ^ = 0, 5=J-j006m7r, Hence \i k^^ k^,,, are the places at which the analytic character of the sum is broken, we have 0=^-1 [8mm^i{/(iti-0)-/(iti+0)}+sin mi-, {/(it,-0)-/(it,+0)H...]. Since this is true for all values of m, the quantities i{-|, it,, ... must be multiples of n ; but 162 THE PROCESSES OP ANALYSIS. [CHAP. VII. there is only one multiple of n in the range 0<g<2ir, namely n- itself. So k^^v, and ^2) ^8> ••• ^o not exist Substituting /r|«s7r in the equation ^-■^— ^oosmfr, we have J-icoswir- [-/(+0)+COSW7r{/(7r-0)-/(»r+0)}+/(2»r-0)]. Since this is true for all values of m, we have i=-^{/(2ir-0)-/(+0)}, and -j=-l{/(^-0)-/(ir+0)}. IT This shews that there is a discontinuity at the point x^nt such that /(t-o)-/(«-+o)-|, , and that /(2,-0)-/(+0)=-|. Example. Find the discontinuities in value of the siun of the series 8ini-^sin22+^sin47-}sin5«+^sin7«-|sin82+^sin I0z+,.., 86. The uniqueneaa of the Fourier expansion. We have seen that it / (z) is a quantity depending on z, and satisfying certain conditions as to finiteness, etc., then the series 00+ 2 (Oro COS m« + 6,n sin m;?), 1 r*' where a^ « - I f(t) cos mt dt (m > 1), 1 r*' 6m=»- / f(t)s]Ximtdt, '^-llj^'^^'- has the sum/(^) when ^ -? ^ 27r, except at the isolated points at which /(z) is discontinuous. The question arises whether any other expansion 00 Co + 2 (Cm COS wu + dm sin mz) of the same form exists, which also represents f{z) in the interval from to 27r ; in other words, whether the Fourier expansion is uniqtte. We may observe that it is certainly possible to have other trigonometrical expansions of (say) the form 00+ 2 (omCOSy+ftnCOSyj 86] FOURIER SERIES. 153 which represent f{z) between and %ir\ for write ;?«2^, and oonsider a function ^(C)) which is such that <^(f)«/(2f) when 0<f<ir, and <^(0-fl^(f) when 7r<f<27r, where g (() is anj other function. Then on expanding ^ (^) in a Fourier expansion of the form 00+ 2 (o^^cosmf+ftncosmf), this expansion represents /(«) when 0<;r < 2fr ; and clearly by choosing the function g (() in di£ferent ways an infinite number of such expansions can be obtained. The question now at issue is, whether other series proceeding in sines and cosines of integral multiples of z exist, which differ from Fourier's expansion and yet represent f{z) between and 29r. If it were possible to have a distinct expansion . 00 /(^) = c^-f 2 (c» COS m* + cijn sin wiir), m-l then on subtracting this iix>m the Fourier expansion we should have an expansion (ao-Co)+ ^ {(a« - Cm) cos T/MT + (6,» - cim) sin 7?Mr} whose sum is zero for all values of z between and 2ir, except possibly a certain finite number of values (namely the discontinuities). The investigation therefore turns on the question whether it is possible for such an expansion as this last to exist. We shall shew that it cannot exist, and that consequently the Fourier expansion is unique*. Let' -Ao^gOo, -^m =■ c^m cos m-? + 6m sin mr (m> 1); and let 2 = -4.0 + Ai + ... + ilm + ••• be a convergent (not necessarily absolutely convergent) series for values of z from to 27r, so that the limit of a^ and hn is zero forn:* oo ; and suppose that (except at certain exceptional points) its sum is zero. Then the series ir ^<r;=sulo^ -ill- -^ -...-— -... converges absolutely and uniformly for this range of values of z, as is seen by comparing it with the series S — . We shall first establish a lemma due to Riemannf , which may be stated thus: * The proof is due to G. Cantor, Journal fUr Math, Lxzn. t Collected Worki, p. 218. 154 THE PROCESSES OF ANALYSIS. [CHAP. YU. The quantity jy F(z + 2a) -¥ F (z - 2a) - 2F(z) ^^ 4? tends to the limit /(z) as a tends to zero, if at z the series X converges to the sumf(z). For the term involving On in J2 is — -r^i (^ ^^^ n (ir -I- 2a) + a« cos n (-«? — 2a) - 2a^ cos nz], On cos nz sin' na or T-z , nV J . ., , ^, ^ . , . I . bn Bin nz sin* na, and similarly the term involving bn is -^ r^ . As F(z) converges absolutely, we can rearrange the order of the terms, and so can write n . . /sinoV . A /sin2a\* Now considering the series 1, we can write say, where z being given, and any small quantity S being assigned at will, we shall have | Cn | < S for values of n > some integer m. Now An ■* «n+i — en for all values of n. Therefore substituting, we have Divide the series on the right-hand side of this equation into three parts, for which respectively (1) l<n<m, (2) m + 1 < n < «, where s is the greatest integer in - , (3) « + 1 < 71. The first part consists of a finite number of terms, each tending to zero as a tends to zero, so the first part is zero. Considering next the second part, the quantities are of the form where <a<w; this quantity decreases as a increases from to TT, so the sum of the moduli of the terms in the second part is less than ^ f/sinmay .sin^a.*] which tends to zero when S tends to zero. 86] FOURIER SERIES. 155 Considering next the third part, we can write the nth term in the. form Ksinn — lay /sinn— laVl «» /• . 7 • • \ or sin* w — 1 a P 1 _ ll _ sin 2n — la sin o ^« a* [—J. n«J *** nV ' 80, as I €n I < S, its modulus is less than Thus the whole sum of the terms in the third part 5 1 sr_i j_ 1 8 r (ic 8 8 8 .8 which is ultimately zero. Therefore the three parts of the infinite series in R are all zero ; and thus 22 '^fi^) in the limit ; which establishes Riemann's lemma. Next, we shall establish another lemma, due to Schwartz *, which may be stated as follows : If a and b are two of the exceptional points, so that between B^a and z^b the series 2 converges to the sum zero, then F{z) is a linear function of z betu)een these values. For assume that a is less than b, and introduce a function 4> (z\ defined by 4,{z)~0^F(z)-F(a)-l-^^{F{b)-F(a)\^-^{s^aHb-z). where ^s 1 and h is any constant Then substituting in the result of Riemann's lemma, we have Therefore (^ + a) + ^ (^ - a) — 2^ (z) is positive when a is very small, whatever be the value of z. Now ^(a)«0 and ^(6) = 0. Also ^(z) is continuous, since F(z) is uniformly convergent, and consequently continuous. Therefore if <f> (z) can be positive between the values a and b of z, it will have a maximum ; let this occur at the value c of z, * Quoted by G. Cantor, Joumaifiir Math. Lxxn. 156 THE PROCESSES OP ANALYSIS. [CHAP. VII. Then when a is small, we have <t>(c + a)-4>(c)<0, and <^(c-o)- <^(c)<0. Adding these relations, we see that the condition just found is violated, and so <f) {z) can not be positive at all within the range. Again, take h small. Choose ^ » j: 1, so choosing the sign that the first term ^ [i^ (x:) — . . .] is positive. Then ^ {z) is clearly positive, if this first term is not zero. But j> {z) is not positive ; and thus we must have Therefore F{z) ia ^ linear function of z, which establishes Schwartz's lemma. We see then that the curve y = F(z) represents a series of straight lines, the beginning and end of each line corresponding to an exceptional point ; and as F{z), being uniformly convergent, is a continuous function of z, these lines must form parts of a polygon. But by Riemann's lemma limit ^i' + '0-f(') F(z-a)-F(z) ^Q Now the first of these fractions gives the inclination of the earlier side of the polygon at a vertex and the second of the later ; therefore the two sides are continuous in direction, so the equation y=^F(z) represents a single line. If then we write F (z) ^cz-^ c\ it follows that c and c' have the same values throughout the range. Thus and therefore A ^ A ^tl I the right-hand side of this equation being periodic, with period 27r. The left-hand side of this equation must therefore be periodic, with period 27r. Thus we have 4^ = 0, c = 0, n h and — c's=ili-H...-H-r cosn^4-~sinn5-H.... nr n' Now the right-hand side of this equation converges uniformly, so we can 86] FOURIER SERIES. 157 multiply the equation by cos nz, sin nz, respectively, and integrate. This gives TT ~ =« - c' I COS nz4^ « 0, ^ J b [^ and TT -4 « — c' I sin nzdz « 0. n« Jo Therefore the a's and b's vanish, so all the coefficients in S vanish ; which establishes the result that the Fourier expansion is unique. Miscellaneous Examples. m 1. Obtain the expansioiis («) 1 — o — ^=H-rooe*+r«coe2f+... , ^ * l-Srcoe^+r" W 5log(l-2rooe«+r^— -rcoB«-jr*co8 2«-5r"ooe3;«-..., (c) tan"** _ — r8in«+5r*8in2r+5r"8in3;«+..., ^ * l-rcoB« 8 3 ' (a) tan"*-= z-=r8m«+s*'^sui3«+rr*8m5*+..., ^ ' 1—1* 3 5 ' and shew that, when | r | < 1, they are oonvergent for all values of t in certain belts parallel to the real axis in the ^plane. 2. Shew that the series - sin — sin^ 8in(n+l) — sin | where all the terms for which i; is a multiple of n are omitted, represents the greatest integer contained in a, for all real values of z between and n. 3. Shew that the expansions ^l0g(2<X)8|j cost - 5C0s2;;+xC0S32... and - log(2 8in rj= -co8«-5 008 2;j-s008 3«... are valid for all real values of «, except mtdtiples of ir. 4. Obtain the expansion • (-l)*^co8fn« , «., /„ '\ . ' / • o . • \ and find the range of values of t for which it is applicable. (Trinity College, 1898.) 158 THE PROCESSES OF ANALYSIS. [chap. VII. 5. Let n be an integer ^ 2, and let XnX^^... x^^i be quantities satisfying the conditions 0<a?i<ar,<...<ar,i_j<l, and write Xq=0, ^n**!* Let 0^, 0], c^t ... c^.i be real arbitrary constants and let a function <f> {a) be defined by the equalities <^(ar)=c^+(Ji+..,+c„ for ar,<a?<it:,+i (i—O, 1, 2, ...n-l), <^(x)-Coforx«Xo, Cm <^(^)-Co+C|+...+c^i+^, for x^Sf Shew that for X^Xn* (*-l, 2, ... n-1), 2 Mai for 0<a?<l, and *jWjt(i2.5+ i «.. where the coefficients a^ and 6^,^ are given by 11-1 ao«2 2 (v(l-.JPr), o,^" 2 (V8in2m9r^r> wiir for m'^ly 1 »-* 6«=--- 2 (v(l-coe2mira?r) for m^l. (Beiger.) 6. Shew that between the values - v and + ir of t the following expansions hold : 3 sin 3s mximz 2 . / sini 2 sin 2s 3 sin 3s \ 2 . COS msB- sin imr (_1_ rncos* fnoos2s moos3s \ 2^'^F^^*'' 2«-m« ^ 3«-m« " "7' -Hg"*^ ^2 / 1 moosg m cos 2s wicos3 s 2*+«? " 3*+wi 7. Obtain the expansions 5 sin/i(f-fmtr) ■-00 s+m*r sin(2n-|-l)s and • cos/*(s+mfr) 2 IB — 00 s+mir Sin I ,sin2nscoti ' oos(2n + l) s sins cos 2ns cot z (2n</i<2n+2) 0A-2nX (2n</i<n+2) 0»-2nX MISC. EXS.] FOURIKB SERIES. 159 lip and q are positive integers, shew that ^ Bin (am -k-p) — ^ 2 i I »~8m — ^- cot^--, ».— ym+jt) ^ ^ ^ ' cos(^+p)?^ * ^^ '^ a IT 2npir .^pn 2 * aa — cos ^ COt-^— . . m— «D ^-m+jt) ^ ^ q 8. Prove that the locus represented by 2 - — I — sinitrsinnyaO is two systems of lines at right angles, dividing the coordinate plane into squares of area nK (Cambridge Mathematical Tripos, Part I., 1895.) 9. If m is an integer, shew that ^ o 1.3.5...(2m-l) fl , »i o . m(m-l) . 2. 4. 6... 2m (2 m + 1 (fn + l)(m+2) m(m-l)(m-2) 1 ■*'(m+l)(i»+2)(nH-3)*^'^'^-7 (a terminating series), ^, 4 2.4.6...(2m-2) fl ^2i»-l ^ r2m-l)(2m-3) ^ . ) ^^'^^■^ 1.3.5...(2m-l) i2-^2iM:i^^-^ (2m-hl^^ (an infinite series). Shew also that , 4/ oos3s.co8 5;s oos7«.oos9« \ and cos 1 - / oosg , ooe3g ^ oosSg cos7i co6 9f \ ' *"irV1.3"^1.3.6 3.5.7"*"6.7.9'"7.9.1l"^*'7' 10. A point moves in a straight line with a velocity shich is initially ti, and which receives constant increments, each equal to ti^ at equal intervals r. Prove that the velocity at any time t after the beginning of the motion is u ^ ut u • 1 . 2mirt - + — +- 2 -am — , and that the distance traversed is trf ,^ . . . Mr VT • 1 2mirt 2;('+") + i2-2,ri,.!,JSS^-T-' 11. Shew that • / .« . sinarir • (-l)*sin(a+2nvir) sin(a + 2tMr9r)» 2 ^ ^— ^ ', It .00 x—n where n is the difference between the real quantity v (supposed not to be an odd multiple of \) and the integer to which v is most nearly equal (Cambridge Mathematical Tripos, Part II., 1896.) 160 THE PBOCfiSSES OF ANALYSIS. [CHAP. VIL 12. Let 9t be an integer > 3, and let ^o, ^|, ^2> <•• be an infinite aet of quantities, which satisfy the conditions, Let X be a real variable, and let s be the greatest integer contained in nx. Shew that when x > 0, 2 ^r= o + 2 {(hnCoa2inirx+b^sin2mnx)f if r is not a multiple of - ; 2 but 2 ^r-o "?+ 2 (aMOOs2fnir^H-6m8in2in9r^, 2 2 masl if r is a multiple of - ; the coefficients a^ and h^ being determined by the formulae fi ^" " •n^ 2 ^,.8in -— (m^ 1), 1^ n— 1 2fitr9r ^"^ — 2 9r<^—— («i>l). (Berger.) 13. Let ^ be a real variable between and 1, and let n be an integer > 5, of the form 4m+ 1, where tn is an int^;er. Let E (a) denote the greatest integer contained in a. Shew that (-1) v»/ + (— 1) v « / = _ + _ 2 —tan cos2m9r^, itx\a not a multiple of - ; but . nnx . ooBnnx 2.2 • 1 ^ 2mfr sin ^ + — ' — ■« - + _ 2 — tan cos 2mirx. if 0? is a multiple of - . (Berger.) 14. Let a; be a real variable between and 1, and let n be an odd number ^ 3. Shew that ( — 1)*«- + — 2 —tan — cos29n9rar, if 0? is not a multiple of - , where < is the greatest integer contained in nx\ but tv MISO. £XS.] FOURIER SERIES. 161 0=-+— 2 -tan — coa2inirx. n n si3| m n 1 if 4? is a multiple of - . (Berger.) 15. Let X denote a real variable between and 1, and let 9i be an integer > 3 ; further, let E(a) be the greatest integer contained in a. Shew that n wi IT m»i f^ n if ^ is not a. multiple of - ; but . ^1 (n-l)(n-2)^l • 1 .mir ^ tu;*— Twr+s" ^ ^ + - 2 — cot - co8 2m»rj?, as on . IT mmi m n 1 if ;p is a multiple of - . (Berger.) ' tit 16. Assuming the possibility of expanding f(x) in a series of the form lAjg sin kx, where 1; is a root of the equation i&co8ai;+6sinai:=0, and the summation is extended to all positive roots of this equation, determine the constants Aj^, (Cambridge Mathematical Tripos, Part I., 1898.) 17. If shew that 6*-l ftsO ^J . C0s4Yr^ . C0s6irJ7 . , ,v. , 2**~*ir** „ , . cos2ir^+— 2^- +— 3s^+... = (-l)"-^-2;jj- V^(x\ ainSiixl""^'"^ I »^^^^^ | _/_n«+i?!!!!:^V (x) 18. If shew that (Cambridge Mathematical Tripos, Part II., 1896.) /(x) = ^Oq + Ox cos a;+ a, cos 2^ + . . . , If shew that a^ss" I f{x)ooanxtsji^x -. If J ^ <^ (j*)=6| sin ^+62sin 2a?+ ... ) b^xM - I <l}{x)axinx tan ^a? — . (Beau.) 19. Prove that the series 2ii»sin - — , 1 <* where ^n^zl sm -— -/(t;) cfv, is equal tof(x) for any value of x lying between and a about which f(x) is continuous. W. A. 11 162 THE PROCESSES OF ANALTSia [CHAP. VIL If /(O), /(a) are the limits of /(«), /{a^^tX when the positive quantity c diminishes to zero, and if /(:r) has sudden increases of value A, k, corresponding to the values a, /3, ... of x, the limit for 7»= ao of nA^ can be written in the form i|/(0)-(-l)./(«)+Acoe^+*co8^ + ...}. Shew that the series sin3a;+- Bin9x+-sin 15:t7+...-2 (sina?+- sin3:F-|--sin 5jf-|-...] ^3^3/ . 1 . K . 1 • ^ 1 • 11 ^ 1 H -jsmd?- r|8ln5ar+=jSm7A•-_-T^smll4?+..,V ha8 the limit -^n* when Xy lying between and 9r, approaches indefinitely near to one or other value, and that it has sudden changes of value -^ir and H-^fr corresponding to the values i n and § n- of :r. (Cambridge Mathematical Tripos, Part I., 1893.) 20. If, for all real values of x, F{x) = AQ+AiCosx+AiCOB2x+A^co&3a;+,,,, then cos(^jF(^)d^= / XF(x)dx, where £r= il^ -I- Jj cos IT + il J cos 4tr + -ij cos 9tt? + . . . , F=4i sin i^+^ij sin 4M;+il3 sin 9tr+ ... , -, X^ ^^ An^^-X^ ira? , ^ 4(2»r)«+a:» 2ir:r . ^=COS -;-+2C03 COS h2C0S — — J- COS + ... Aw Aw w Aw w 4(2n»r)«+a:» ^nirx + 2 COS -^ — 7 cos . Aw w Prove these formulae, and thence deduce the result {U+ V) (^y-l^(0)+^Mco8^^'?+/'(2«-)oo8 ?*;"+... 2ir where to^-r- ^ k being a positive integer. When k is even, the last term of each series involves F{ikw) and is to be multiplied by ^; when k is uneven, the last term involves F{i{k-l)io), (Cambridge Mathematical Tripos, Part II., 1896.) CHAPTER Vni. Asymptotic Expansions. 87. Simple example of an asymptotic expansion. Consider the function • el^Ht J 3 t where x is real and positive, and the path of integration is the real axis in the ^-plane. Integrating by parts, we have and by repeated integration by parts, we obtain ^/ \ 1 1.2! . (-!)»-' (»-l)! / ,w , r«^*d« In connexion with the function f{x\ we therefore consider the series 1 1^2! ^ (-!).-.(„ -1)1 X oc^ a? '*' x^ We shall denote this series by S, and shall write 1 12! (-l)nn! _ X ^■*"a;» •••"^ (^"^^ "'^~* n-1 The ratio of the nth term of the series & to the (n — l)th term is : ^ ' X for values of n greater than 1 + a?, this is greater than unity. The series 8 is iherefo7'e divergent for aU values of x. In spite of this, however, the series can under certain circumstances be used for the calculation of f{x); this can be seen in the following way. 11—2 164 THE PROCESSES OF ANALYSIS. [CHAP. YIII. Take any definite value for the number n, and calculate the value of Sn^ We have /(^)-Sfn-(-l)«+'(«+l)!/J'^f. and therefore \/(x)-8n\=(n + lV.( J i — ^ , since e*^ < 1 and t is positive^ < ^n+i- For values of x which are suflBciently large, the right-hand member of this equation is very small. Thus if we take x > 2n, we have 1 |/(^)-^n|< 2«+i n 1 > which for large values of w is very small. It follows therefore that the value- of the fwaction f(x) can be calcukUed with great accuracy for large values of X, by talcing the sum of a finite number of terms of the series 8. The series is on this account said to be an asymptotic expansion of the- , function f{x). The precise definition of an asymptotic expansion will now^ be given. 88. Definition of an asymptotic expansion, A divergent series -^0 + — H"-^ +.••• + ^ + •••» in which the sum of the (w + 1) first terms is Sn, is said to be an asymptotic expansion of a function f{x), if the expression af^ \f{^) - Sn] tends to zero as X (supposed for the present to be real and positive) increases indefinitely- When this is the case, if a? is sufficiently great, we have where € is very small ; and the error — committed in taking for f{x) the- (n-f 1) first terms of the series is very small. This error is in fact infini- tesimal compared with the error committed in taking for f{x) the n first, terms of the series : for this latter error is of" ' and € is in general infinitely small compared with iln + €. 88,89] ASYMPTOTIC EXPANSIONS. 165 The definition which has just been given is due to Poincar^*. Special asymptotic expansions had, however, been discovered and used in the eighteenth century by Stirling, Maclaurin and Euler. Asymptotic expan- sions are of great importance in the theory of Linear Differential Equations, and in Dynamical Astronomy ; these applications are, however, outside the scope of the present work, and for them reference may be made to Schle- singer 8 Handbuch der Theorie der linearen Differentialgteichungen, and the second volume of Poiucar6's Les M4thode8 Nouvdles de la Micanique^ Cileste. The example discussed in the preceding article clearly satisfies the definition just given : for n\ and the right-hand member of this equation tends to zero as x tends to infinity. The term "asymptotic expansion" is sometimes used in a somewhat wider sense ; if F, ^, and / are three functions of a?, and if a series X or is the as3anptotic expansion of the function F ' we can say that the series ^ „. FA^ FA^ X as" is an asymptotic expansion of the function J, For the sake of simplicity, we shall consider asymptotic expansions only in connexion with real positive values of the argument. The theory for complex values of the argiunent may be discussed by an extension of the analysis. 89. Another example of an asymptotic expansion. As a second example, consider the function f{x), represented by the series .(1), where c is a positive constant less than unity. The ratio of the Arth terra of this series to the {k — l)th is less than unity when k is large, except when a? is a negative integer, and conse- ♦ Acta Mathematical vin. (1886), pp. 295—344. 166 THE PROCESSES OF ANALYSIS. [CHAP. VIII. quently the series converges for all values of x except negative integral values. We shall confine our attention to positive values of x. We have, when x>k, 1 I k k" 1^ h = = + -^--.+ -7-.... X + k X (/^ s? SK^ a^ If, therefore, it were allowable to expand each fraction , in this way, X T" fu and to rearrange the series (1) according to descending powers of a?, we should obtain the series / t4"-^-4'^ w- 00 where ili= 2 c*; il3 = — 2 A;c*, etc. But this procedure is not legitimate, and in fact the series (2) diverges. We can, however, shew that the series (2) is an asymptotic expansion of /(a?), which will enable us to calculate /(a?) for large values of x. Forlet' s„ = =^ + ^» + ...+^'. X a^ af^^^ and ^{/(^)-£f„} = LjL2_ 2 ^. Now 2 r is finite, and so when x is infinitely great the right-hand t— 1 X -h K member is infinitesimal. Therefore a?"{/(a?) — /Sn} tends to zero when x tends to infinity; and so the series (2) is an asymptotic expansion of f(x). Example. l{f(x)= I e^^^dt^ where x is supposed to be real and positive and the path of integration is real, prove that the divergent series \ 1 1.3 1.3.5 is the asymptotic expansion otf(x). 90] ASYMPTOTIC EXPANSIONS. 167 90. Multiplication of asymptotic expansions. We shall now shew that two asymptotic expansions can be multiplied together in the same way as ordinary sehes, the result being a new asymptotic expansion. For suppose that X a^ '" af^ •^0 '• "T" "^ ^^"T •••4" _„■ > ••• are asymptotic expansions representing functions J{x) and J'{x) respectively, and let 8^ and 8n be the sums of their (n + 1) first terms ; so that Limit a?« (/ - 8n) = 0^ (1). Limita;»(J-'-/S„') = ap=300 , Form the product of the two series in the ordinary way ; let it be and let Sn be the sum of its n first terms. As 8n, 8n and 2» are simply polynomials in - , we have clearly X Limit «»(flf„S„'-S„) = (2). gmto Now by (1), we can write '^ = ^« + ^' where Limit e = 0, Limit e' = 0. ee' Then a!» (J J' - S„ 5«') = Sn'e + /S„€' + ^: . X The terms in the right-hand member tend to zero as x tends to infinity. Hence Limit a?«(JJ'-Sn/Sfn')=0..... (3). «B00 168 THE PROCESSES OF ANALYSIS. [CHAP. VIIL From (2) and (3) we have Limit a^ {J J' - 2n) = 0, X=oo and therefore the series X or is the asymptotic expansion of the function JJ\ 91. Integration of asymptotic expansions. We shall now shew that it is permissible to integrate an asymptotic expansion term by term, the resulting series being the asymptotic expansion of the function represented by the original series. For let the series represent the function J(x) asymptotically, and let Sn denote the sum Then, however small a real positive constant quantity € may be taken, it is possible to choose x so large that and therefore J(<')-Sn\<^, \r J{x)dx-{ Sndx\^r\J{x)-8n\dx \j 9 J X I J » ^(n-l)«»-»' and therefore the integrated series a, ^2<r»+-+(n-l)«»-> + - is the asymptotic expansion of the function J{x) dx. J a X On the other hand, it is not in general permissible to di£ferentiate an asymptotic expansion. 92. Uniqueness of an asymptotic expansion. A question naturally suggests itself, as to whether a given series can be the asymptotic expansion of several distinct functions. The answer to this 91, 92] ASYMPTOTIC EXPANSIONa 169 is in the affirmative. To shew this, we first observe that there are functions L{x) which are represented asymptotically by a series all of whose terms are zero, i.e. functions such that Limit x^L {x) = 0, ivhatever n may be, when a; (supposed to be real and positive) increases indefinitely. The function e~* is in fact such a function. The asymptotic •expansion of a function J{x) is therefore also the a83anptotic expansion of J{x)-¥L{x). On the other hand, a function cannot be represented by more than one -distinct asymptotic expansion for real positive values of x ; for if X a^ -4o+— + —'+.. . -and jBo + — +-?+... X a^ «re two asymptotic expansions of the same function, then U^i^(A4" + ...4--A-4....-|.).„, which can only be if ul© = 5© ; -^i » -Bi, etc. Important examples of asymptotic expansions will be discussed later, in connexion with the Gkunma-function and the Bessel functions. Miscellaneous Examples. 1. Shew that the series 1 1 ' 2 ' 3 ' - + —+ — + — -I- is the asymptotic expansion of the function / dt t when X is real and positive. 2. Discuss the representation of the function (where x is supposed real and positive, and <^ is an arbitrary function of its argument) by means of the series 170 THE PROCESSES OF ANALYSIS [CHAP. VDI. Shew that in certain cases (e.g. (f)(t)^e^ the series is absolutely convergent, and represents f{x) for large positive values of j? ; but that in certain other cases the series is the asymptotic expansion o{f(x). 3. Shew that the divergent series I a-l (a -l)(a-2) is the asymptotic ezpemsion of the function «-• r* log^j, for large positive values of z, 4. Shew that the function /(.)=/; (iog«-.iog(^)}^- 6-»» has the asymptotic expansion " where By^B^y ... are Bemoulirs numbers. Shew also that/(j;) can be developed as an absolutely convergent series of the form 5. Shew that the function has the asymptotic expansion 1.3.*.(2n-3) 00 PAET II. THE TRANSCENDENTAL FUNCTIONS. CHAPTER IX. The gamma-function. 93. Definition of the Oamma-function : Euler*8 form. Consider tlie infinite product -n n This product clearly diverges if -e is a negative integer, for then one of the denominator-factors vanishes. If z is not a negative integer, the product will (§ 23) be absolutely convergent, provided the series Jj.log(l + l)-log(l + i). is absolutely convergent ; but since when n is large we have the terras of this series ultimately bear a finite value to the terms of the series and therefore to the terms of the series 2 — , which is absolutely convergent. The infinite product is therefore absolutely convergent for all values of z, except negative integral values. 174 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. This product may be regarded as the definition of a new function of the variable z\ we shall call it the Oammorfunction, and denote it by r(^), so that r{z)=-u i.('-S" n This form of the function was fii-st given by Euler ; but the notation r(^) is due to Legendre, who applied it in 1814 to an integral which will presently be discussed, and which represents the Gamma-function in some case& Excunple. Prove that r(;?)= Limit -r-r -ryr/— — ^T^'• (Gauss.) •«=«, «(«+I)...(«+n-l) 94. The Weierstrassian form for the Oamrnxi-fmiction. Another form of the Gamma-function can be obtained as follows: We have , n+l v{z)^- n^ *»-! 1 + 1 n = iLimit «*><«'"'+« fi ^ Z fHsoo "-('^9 = - Limit 6'{"*<~+"-'-«---»}n -21-. 1+- n Now ! + _+.. .+--.-log(m+l)= t (--log n+1 n = 1 ni-±-)i^ Joln-iw(n-fa?)j Now the series 2 —. r is absolutely and uniformly convergent for real values of a? between and 1, as is seen by comparing it with the series * 1 n=l ^ 94] THE GAMMA-FUNCTION. 175 hence as m increases, the right-hand member of this equation tends to the limit 1 ( 00 io(n=iw(n+"^ which is finite, since the range of integration is finite and the sum of the w series 2 —7 r is finite. This limit is known as Euler's constant, and we ii=in(n+a?) shall denote it by 7, Its numerical value is 0-5772157. ... Thus Limit ]i + h+... + log(m + l))- = 7, z 1 * 6 and 80 r (^) = - e-^' II or ^ n=i 2 , f^ ' n }-^zey'U |fl + ^)^"*l W n=l (V n) J This form (due to Weierstrass) shews that yr7-\ is a regular function of z for all values of z. Example 1. Prove that 1^(1)= -y, where y is Euler's constant. For differentiating logarithmically the equation and putting z=\ after the differentiations have been performed, we have -r'(i)=i+y+i(^^-l). or r'(l)«-y. Example 2. Shew that 2 3 n Jo » and hence that Euler's constant y is given by ' Jo « Example 3. Shew that the infinite product 176 TBANSCENDENTAL FUNCTIONS. [CHAP. IX, has the value For nil — )««= n e* i»»i \ z-¥nJ n=i n+z *^* {n+z)e'H 5 (i+izf) e n n fl+-^tf-^ The numerator of this expression is Weierstrass' form of 1 («-j?)6>^-'>r(a-a?)' and the denominator is 1 ze^'riz) Therefore the given expression has the value e^zr (z) 96. The difference'equaiion satisfied by the Gamma-function, We shall now shew that the function r(^) satisfies the difference- equation r(^ + i) = ^r(^). We have 1 «(^ + 9 r(^+i) = -^n — 2i_ 1 + n IVn-^l z-^-ln^i n-^^ + l n 00 1 + 71/ il V W/ u- — —= n n+1 n This is one of the most characteristic properties of the Gamma-function. It follows that if 2r is a positive integer, we have r(z) = (^-i)! 96,96] THE GAMMA-FUNCTION. 177 Example, Prove that 111' e r(r+i) r(z+2) ' r(«+3) \z l!«+l'*"2!«+2 Sl^+a"*" •••J • For consider the quantity 1 1 1 ^'*"«(i5+l)'*"^(«+l)(z+2) **■•••• This can be expressed as the sum of a number of partial fractions, in the form z ^«+l^**'^«+n^*'" To find the coefficients a, multiply by {z-\-n) and put «= -n ; we thus obtain 1 fll 1 )(")*« ^ ^"'(-i)"7i!r'*'i'*'r2"*"r70"^ "J "~w^ * Therefore z But 2^«(2+l)^z(«+l)(z+2)^- ^\2 (0+1)1! (^+2) 2! -J 1 r(«+n+l) «(z+i)...(«+«) r(2) whence the required result follows. 96. Evaluation of a general cla^s of infinite products. By means of the Gamma-function, it is possible to evaluate the general class of infinite products of the form 00 n Un, where v^ is any rational function of its index n. For resolving Un into its factors with respect to w, we can write the infinite product in the form Yi j Q/t - ai)(n- g,) ... (n-aj „«i ( (n-6i) ...(n-6j) In order that this product may converge, it is clearly necessary that the number of factors in the numerator may be the same as the number of factors in the denominator; for otherwise the general term of the product would not tend to the value unity as n tends to infinity. w. A, 12 178 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. We have therefore A: = /, and can write (denoting the product by P) n-il(n-6i) ... (n-6fc)j For large values of n, this general term can be expanded in the form ('-3-('-?)('-r-('-r' or 1 h terms m — r + . . . . n n' In order that the infinite product may be absolutely convergent, it is therefore further necessary that Oi + ... + ajb — 6i — . . . — 6t = 0. We can therefore introduce a fiwtor e ^ in the general term of the product, without altering its value ; and we thus have But n j(l--)e"U „/ - .^^ . Therefore P= ^^^^dhlM (=*«) •_-.l»£(zi_M a formula which expresses the general infinite product P in terms of the Gamma-function. Example 1. Prove that •J" «(«_-H5>+f) ^r(a+i)^r(6+i) ,=i<a+«)(6 + *) "r(a + 6 + l) * Example 2. Shew that ^(i-g)^i-g)...-{-r(--:»;^)r(-a^)...r(-a-iJ)}-S where 2ir . . 2ir a=cos hi Sin — . n n \ 97, 98] THE GAMMA-rUNCrriON. 179 97. Conneodon between the Gammorfunction and the circular functiovs. We now proceed to establish another of the characteristic properties of the Gamma-function, expressed by the equation r(z)r(l-z)=^ IT SUITTZ We have 2(i—Z) „.i (•^3 ]^\«+l— « W-f 1 1 • n 1 • n Zil-Z) n^i =i n ^ (•-9('-.-Tl) '■(-5) TT Sin TTZ which is the result stated. Corollary. If we assign to z the special value g, this formula gives |rg)('=..orr(i) = .* 98. The multiplication-theorem of Oauss and Legendre. We shall next obtain the result »-i r(^)r (^ + 1) r [z+fj ... r (^ +^) = r (n^)(2,r) * «*-». n ni; For let 0(^) = r(^)r(^ + ^)...r(;r+ n-l w Then <l>(z nF (n^) ^rur+nf (^ + 1) F (^ + 1 + i) ... F (^+ 1 + ^^) nF (n-2 + w) (m + n - 1) (nz + » — 2) ... (nz) ^W 12—2 180 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. It follows from this that ^ (2) is a one- valued function of 2, with the period unity ; and <f> (z) has no singularities when the real part of ^ is positive, since =-7 — r is ever3rwhere regular ; it has therefore no singularity for any value of z, and so by Liouville's theorem (§ 47) it is a constant. Thus <j) (z) is equal to the value which it has when ^ = - ; which gives The«f.„ *F^- {r (1) r (I - 1)) {r (?) r (1 - 1)} (by § 97) = 1:11 ^ (2^^!: Thus <l>(z) = . TT . 27r . (n — l)7r n sm - sm ... sin ^ — n n n «-i or r(^)r(^+^) ...r(^+'^) = r(n^)n*-~«(27r) n-1 2 Example. If shew that Binp,nq)=n-'^ it(g,g)iK2g,g^ ... ^K>^- l)g, g} ' 99. Expansions for the logarithmic derivates of the Gammorfunction. We have {r (z + !)}-» = ^^n (1 -I- ^ €~\ DiflFerentiating logarithmically, this gives dlogr(^ + l) _ f ^^ 1 1 \ dz '^■^^(l(^+l)"^"2(^+2)"'"3(^ + 3)"^"y Also iogr(^ + i) = iog^ + iogr(^), so ^iogr(.+i)-U^iogr(.). 99, 100] THE GAMMA-FUNCnON. 181 Therefore ^iogr(.) = l+*iogr(.+i) 1 d_( z + 1) "^'2(^ + 2) ■*■••• _ 1 1 1 "■^«'*"(^ + l)>"^(^ + 2)«^-- These expansions are occasionally used in applications of the theory. 100. Heine's expression ofT{z)asa contour integral. It has long been recognised that the Oamma-fiinction is intimately connected with the theory of a large and important group of definite integrals ; and in fact the function has frequently been defined by means of a definite integral. We now proceed to consider various definite integrals in this connexion, the most general of which is due to Heine and can be obtained in the following way. We have /, IV i^('-3" r(^) = - n m 1 ^- 7n Now if we express - 11 in partial fractions, we obtain i n — = 1 (-i)" — — =— ^ m»i z-^m mao tn ! (n — m) \ z + m Consider now the function (- ^)-'. This, when a is a complex quantity, may be defined as being equiva- lent to r«-i)iog(-») Now the logarithmic function is many-valued, since the value of the function log (— w) is increased or decreased by 27ri when the variable x describes a simple circuit round the point a? = 0. In order that the function (- a?)*"* may have a unique value, we have therefore to select one of the different determinations of log (-a;): and this may be done in the following way. We first make the stipulation that the variable x is not to cross the real axis at any point on the positive side of the origin ; this prevents x from making circuits round the origin, and so makes each of the determinations of log (— x) a single-valued function. Then we select, fix>m these determinations, that one which makes log (— x) real when ^ is a real negative quantity. The value of log (— x) being thus uniquely defined for every value of x, it follows that the value of (—a:)*"* is likewise uniquely defined. 182 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. With these presuppositions, if (7 be any simple contour enclosing the origin and cutting the real axis in the point x^l,we have clearly f , , , , r (-a?)»l 6^ c-'*^ 2tsin7ra Jc L(7 « J a a a Therefore iniLl^=,-.i-j(i+i)...(i+i)ri f (-rrv-.y ^m«i 1 , ^ 2sm7r^(V 1/ V w/j ««oJ(7 \mj^ ' +m-i^ 1+- (n + iyf (- a?y-i (1 - a?)** (ic. 2 sin TT-er ^ J c Writing y = ti^ in this equality, we obtain IV 1 liiL!j2l = _^ m where 2) denotes any simple contour in the plane of the complex variable y^ enclosing the point y = 0, and cutting the real axis in the point y«n. If now we make n increase without limit, we have r (z) = ^ — /"(- yY^'e-ydy, ^ ^ 2 sin irzj ^ ^' ^ where the integral is taken along a curve commencing at positive infinity, circulating round the origin in the counter-clockwise direction, and returning to positive infinity again ; and in the integrand we must take (— yy^^ as equivalent to e}*^^ ^^ ^■^^ where the real value of log (— y) is to be taken when y is negative, and the logarithm is rendered one- valued by the stipulation that the variable is not to cross the real axis at any point on the positive side of the origin. Since r(5)r(l--^) = -r IT SmTT^ this result can be written in the form 77)-2^/(-y>"'^"^^y- V{z) This theorem is valid for all values of z — in contrast to that found in the next article, which is true only for restricted values of the variable. Example 1. Bourgvs^s expressions for the Oamma-function, By a slight extension of the above proof, it is seen that 101] THE GAMMA'FUNCTION. 183 where the path of integration is restricted only to contain the origin and to be extended indefinitely at both ends in the direction of the negative part of the real axis ; the contour need not be closed. Take then as contoui* two lines inclined at an angle a to the axis of Xy passing through the origin, and a small circle round the origin. The integral round the small circle is zero when z has its real part comprised between and 1. The integration along the two lines gives the result ^^'^^-^ rp*-V«-*sin(p8in«+^)c;p, which can be written in the form Tiz)^-. \-. rp'-^ef'^^'' Bin (p-hza) dp, ^ ' sm zir (sm a)* J This formula is true for all values of a which are not less than -^ . Taking a equal to ir, we have the result r(«)=| p'-^e-Pdp, EaximpU 2. By taking for contour of integration a parabola with the origin as focus, shew that r(*)" 2,-ig-^^^ / «"**'(l+^*"*<5<>8[(2«-l)tan-ia?+^]cir. (Bourguet) 101. Expression of T {z) as a definite integral, whose path of integration is real. We have, by the result of the preceding article. r (z) = =r^ { (5-V+ <«-« i<« <-y' dy. Take a path ABODE, commencing at the positive infinitely distant extremity of the real axis (which considered as initial point we denote by A\ proceeding close to the real axis until it arrives at the neighbourhood of the origin, describing a small circle BCD round the origin, and returning, close to the real axis, to positive inBuity again (which, considered as terminal point, we denote by E). With the conventions that have been made, the integral along the part AB of the path becomes i f^ I e^+(*-i)logy-t>(z-i)^y 2 8m7r^;^ ^ in which log y is supposed to have its real determination. The part of the integral due to the small circle BCD is easily seen to be zero if the real part of z is positive. For the part of the integral due to DC, we have 2 8in7rjrJo 184 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. Thus 28m TT^r Jo or Jo This integral is called the Eulerian Integral of the Second Kind, It is frequently given as the definition of the Gamma-function : but for this purpose it is unsuited, since the integral exists only when the real part of g is positive. Example 1. Prove that when z is positive Example 2. Prove that Example 3. Prove that * e'^x^'^dx ^X^'^dx^^\ (^+1)- • (^+2)* ' («+3)* • r(«);o «*-! ' 102. Extension of the definite-integral expression to the case in which the argument of the Gamma-function is negative. The formula of the last article is no longer applicable when the argument z is negative. Saalschiitz has shewn however that, for negative arguments, an analogous theorem exists. This can be obtained in the following way. Consider the function r, (^)= JV> (^--1 +^- ^,+ ... + (- iy^'^)dx, where ^ is a negative number lying between the negative integers — k and -(A: + l). By partial integration we have, when ^ < — 1, r.w-[f («-■-.+.- ?,+...+(-!)"■ i;)] 09^ 00 102, 103] THE GAMMA-FUNCTION. 185 The terms in the left-hand member which are not under the integral sign vanish, since (z + k) is negative and (-er + A + 1) is positive : so we have The same proof applies when z lies between and — 1, and leads to the result T(z+l)=^zr,{z) {0>z>-l). The last equation shews that, between the values and — 1 of ^, r,(z)-r(5). The preceding equation then shews that Fi (z) is the same as F (z) for all negative values of z less than — 1. Thus for all negative values of z, we have SaalschUtz's result F(^)=jV^(6-«-l+a:-|^j.h.:.+(-l)*+^|^)^^ where k is the integer next less than — z. Example. If a fimction P (ji)he such that for poeitiye values of fi we have and if for negative values of fi we define P^ (ja) by the equation where k is the integer next less than -^, shew that 1 1 .,-.., 1 AW=PW-- + fi-(;^)-... +(-!)*- ^7(;^ (Saalschiitz.) 103. 0atL88* eaypression of the logarithmic derivate of the Gammorfunction as a definite integral. We shall next express the function -7- logF(r) as a definite integral, where z is supposed to be a positive real quantity. 1 r* We have - = / er^dx, 8 Jo Therefore loe « = I -de— 1 dx. ^ Jis Jo on 186 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. Thus we have I e"*fi^"* log sds « I er's^^ds j — dx Jo ^ K Jo Jo ) r<.).rw/;^|<r--,ji^}. This equation is due to Dirichlet. Writing 1 -♦- a? = e* in the second term of the integral, and a? = ^ in the first term, we have which is Qauss' expression of -r- log r(^) as a definite integral. ^l0gr(«)- / I— Irr-^J rf^ . (Gauss.) Example 1. Prove that dz Example 2. Prove that d. ^,, , 1 P . r«(l-a) a(l-a)(2-a)^ "1 ^■-/•■{q--} 104. Binet*8 expression of log T (z) in terms of a definite integral. Binet* has given an expression for log r(^), which is of great importance as shewing the way in which log F (z) increases as z becomes very large ; his result will be used later in the derivation of the asymptotic expansion of r {z). We have by the last article (z being supposed real and positive) r changing -^ to ^ + 1, we have 5"«I'(' + ')-/"(t-*^)*- Now f er^dt^^, Jo ^ * JawmaX de Vie. Polyt xvi. (1S39), pp. 128—343. 104] and THE OAMUA-FUNCnON. 187 /, — (e « t Jo Jo Jo y = log z. Therefore |iogr(.-Hi)=^+iog.+/;d*{-?^^-?:^ -< — /?-t« ,-e -te ^ e ■^ t 6«-l or Integrate for z between the limits 1 and z ; so iogr(z+i).|iog^+^(iog^-i)+j"J{^-l+l i<«r(.)-(.-l)iog.-.+/;{j^,-l+^}r? t dt dt *'-fA^A*-^r'' ('> The first of these integrals can be otherwise expressed in the following way. We have* -l"^2 t Jo e^'^-1 ' Multipljong both sides of this equation by e~^ dt and integrating with respect to t from zero to infinity, we have ^Jo (u' + udu (w*+i>*)(e^»-l)* Integrating this equation from j> » ^ to /> = oo , we have *• i:?*(i^i4-i)- tan" ■ © dt 6^-1 (2). * A proof of this equation can be foand by making k infinite in the equation 2 -rs-i — 0=2 S I «-*•»« sin (ttt) dt*. 188 TRANSCENDENTAL FUNCnONS. Thus equation (1) becomes logr{z)^(^z~^logZ'Z'^2 tan"" ■©"• e«'»-l ^^ Jo V-l t^2 --dt t Now write z^^^ equation (1): since r© "», we obtain 1 , 1 r* ( 1 11 6 -i< t dt -rA^.-w^T^ t Write ^ for t in the last integral. Thus or = = g log IT 5 log IT _i_r{_i 1 r 2 Jo V-1 ^^^I'^t 1 f» f -1 e-f" 2 Jo te'-l''" « , dt, e-^) dt t Adding this to equation (3), we obtain ' 1\ logr(«) = lir-2Jlog«-« + 2 [chap. IX. tan~M-)du , , .(3). +Jo « v-i t (^-i^ t 2} ^*^- The last integral is rdt(l , tr* -e-»«l t or /ft J ^ JO t or I log X dx, * This ftrtifioe it dae to Pringsheim, MatK Ann, xxxi. 105] THE GAMMA-FUNCTION. 18^ or . 'l^ogl-^l. Substituting this in equation (4), we obtain logr(^) = ^^-2Jlog^-2: + 2log(27r) g2iru_l This is Binet's formula for log r(z)] ss z increases indefinitely, the last integral diminishes indefinitely, and so the remaining terms furnish an approximate expression for logr(^:) when z is large. Example, Prove that log r(«)=(«-i) log«-«+i log (2ir)+ J(«), where J (z) is given by the absolutely convergent series •^ ('^"* |m^ l"*" 2 (2r+l)V+2)'*' 3 (^+l)(« + 2)(«+3) ■*"•••/' in which ^i~J> ^2~J> ^"fo> ^4"*Tcr> and generally c^z^j (x+l)(a?+2)...(a?+n-l)(2x-l)a7ctr. (Binet) 106. The Eulerian Integral of the First Kind, The name Eulerian Integral of the First Kiixd was given by Binet to the- integral jB(p, ?)= [ x^^ {I - x)^-^ dx, Jo which was first studied by Euler and Legendre. In this integral, the real parts of p and q are supposed to be positive ; and x^~^, (1 — x)^^ are to be understood to mean those values of e^^^^**^* and e<^*^**^^""*^ which correspond to the real determinations of the logarithms. With these stipulations, it is easily seen that the integral exists, since the infinity of the integrand is of less than the first order at the two extremities, of the path of integration. We have, on writing (1 - a?) for x, B{p.q)=B{q,p). Also or It 190 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. Also Jo Jo = Bip+l,q) + B{p,q+l). Combining these results we obtain the formula Example 1. Prove that if n is a positive integer, r>/ i-iv * 1.2... tl ^(^'''+l)-p(^+l)...(p+n)- Example 2. Prove that 106. Expression of the Evlerian Integral of the first kind in terms of the Oamma-function, We shall now establish the important theorem ^. ^ r(m)r(n) B (m, n) = 't^^ — -/ . ^ * ^ r (m + n) To prove this, we have /•OO /•OO r (m) r (n) = I e-%^*-^da; x | e-yy«-^ dy JO ./o (writing a^ for a:, and y" for y) Jo Jo 4 / e-^'^-^^^af^^y^-^dxdy J J (writing r cos ^ for x, and r sin for y) = 4 I f e-*«r*("*+n)-i cos*^^ sin«^-^ ^ dr dtf ^0 Jo = r (m + n) 2 I cos«^» ^ sin*^i ^ d^ (putting cos* 0^u) = r (m + n) -B (m, n). This result connects the Eulerian Integral of the first kind with the Gamma-function. 106 — 108] THE QAMMA-FUNCnON. 191 Example. Prove that (Cambridge Mathematical Tripos, Part I., 1894.) 107. Evahiation of trigonometrical integrals in terms of the Oamntar- function. We can now evaluate the integral w •a I cos*'*~'a?8in**"^a:da?, Jo where m and n are not restricted to be integral, but have their real parts positive. For writing sin* a?= ^, we have r* 1 P ^-1 5.1 1 r/^ n\ - 2 , (^.) The well-known elementary formulae for the case in which m and n are integers can be at once derived from this Example. Prove that when | iE: | < 1, pcoS^8in*^flW V 2 y \ 2 y P cos*»-"*^(f ^ jo O^siE^^jT" ^- /m+n+l\ -/<> (^.^^i^,^)!^- (Trinity College Examination, 1898.) 108. DirichleSs multiple integrals. We shall now shew how the integral may be reduced to a simple integral, where / is an arbitrary function of its argument, and the integration is extended over all the systems of positive values of the variables x, y, z, which satisfy the inequality 192 TRANSCENDENTAL FUNCTIONS. [chap. Et. Write 111 a?«(L»i*, y^hyf, z^cz{'. Then the integral takes the form / = a^b^C a/87 - jjJA^ + yi + ^) x^^^-^yi^^-^z^^-^dx^dy^dzu where the integration is now taken over all the systems of positive values of the variables ah, yi, ^1, which satisfy the inequality ^ + yi+'«i^i. Now let /i = ^+yi + ^i-f=0, be three equations defining new variables f, 1;, f. -1 a(/i,/2,/s) Then 9(^1, yi, ^1) -^ -1;? 1 1 1 1 1 1 =fV The field of integration is clearly such that the new variables f , 1;, f, each vary from to 1. Thus / = ^^^^^ C f ^ f V(f ) P'+^»+''»-^ (1 - ^)^'-»i7«i-^'-^ (1 - O^^-'^-'d^dvd^ otp7 JoJoJo a^7 -50),, 9. + r,)5(g.. r,)f /(f)P-^''+"-'df Jo "a^7 \a i8 7> The multiple integral is reduced to a simple integral. It is easily seen that this method of evaluation can be applied to multiple integrals of a similar form in any number of variables. 109] THE GAMMA-FUNCTION. 193 Example 1. Shew that the moment of inertia of a homogeneous ellipsoid of unit density, taken about the axis of «, is -i(a«+i^«)fra5c, where a, 6, c are the semi-axes. Example 2. Shew that the area of the epicycloid ^ -)-y^s= ^ is |ir^. Note, Dirichlet's integrals can also be evaluated by performing on the variables the substitution a?i«sf^ sin* $1 sin* ^„ yi «= r* sin* ^j cos* ^j, «i»!r*cos*^i, leading to the same result as above ; in the case of an integral with n variables the corresponding substitution would be a?i «r* sin* $1 sin* 6^ ••• sin* ^»- j, etc. 109. The asymptotic expansion of the logarithm of the Oamma-function {Stirling's series). We now proceed to 6nd an expansion which asymptotically (§ 88) represents the function logr(i^), and is actually used in the calculation of the Qamma-function. For simplicity, we shall consider only real positive values of the argument z. For a proof and discussion of the expansion when z has complex values the student is referred to a memoir by Stieltjes*. From Binet's expression for log F (z) (§ 104), we have logr(^) = (^-5)log^-^ + |log(27r) + <^(^), tan"' - ax where ^(^) = 2/ ^^^ r) = 2J * A Now tan-- = ----+---... (-1 y»-> ^-^ (-1) ^ r* t^dt '*"(2n-l)~^^-i"^ z^^^ Jo t^ + z* Substituting this in the integral, and remembering that a?**"^ dx Br. / e»»*-l 4n' * Lumvt2Ze*< J<mmal (4), v. pp. 425—444 (1889). W. A. 13 194 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. where jBj, fij, ... are the BernouUian numbers, we have Let us now find approximately the magnitude of the last term when z is very large. r* dx (^ t^dt The quantity j^__J^__^ is less than '''' ^1n+l)^«Jo or 4(n4-l)(2n+l)z«* If now any value of n be taken, it is clear that this quantity can be made as small as we please by taking z sufficiently large. It ollows that the quantity |<PV^; r^i 2r(2r-l)^»-iJ can be made as small as we please by taking a sufficiently large value for z ; and therefore (§ 88) the series -r l,2.z 3.4.^ ■ 5,Q.z' is the asymptotic expansion of the function <f> (z) for large real positive values of z. We see therefore that the series l\i If /« N 5 (-ly-^Br 1 ar— 1 is the asymptotic expansion of the function logr(z) for large real positive values of z. This is generally known as Stirling's series, 110. Asymptotic expansion of the Oamma-function, Forming the exponentials of both members of the equation just found, we have S B S r (z) = e-^z'-i(2v)i^''~^^^^^'"" , 110] THE GAMMA-FUNCTION. 195 or r (^) = «--^-i (2,r)* |l + ^ + § + . . .1 , where Cj = z-— ^ , C, = -^ , etc. Substituting the numerical values of the Bemoullian numbers, the formula becomes n,z)-e z- \''^}'y-+i2z^2{12zy 30(12^)' 120(12^)* "'j' This is the asymptotic expansion of the Oammorfunction, In conjunction with the formula r(l -{-z)^ zT{z), it is very useful for the purpose of com- puting the numerical value of the function. Tables of the function log r {z\ correct to 12 decimal places, for values of t between 1 and 2, were constructed in this way by Legendre, and published in his Exercices de Calcul Integral, Tome il p. 85, in 1817. Miscellaneous Examples. 1. Shew that <-.>(-l)('-3)(-l)--7(:;^- (Trinity College Examination, 1897.) 2. If o-n be the sum of the n first terms of a divergent series - + - + - + ..., «i Oa «8 shew that the series 1 1 1 Ojo-i OfO-f a^^ is divergent. If the squares of the terms of the latter series form a convergent series, shew that a function O (l+z) can be defined by the equation 6^(1+^)= Limit <rn' and shew that tf(T+7)-^°.{(i+i-J «■'••'-}> where e is a constant. (Ceearo.) 13—2 196 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. 3. Prove that dz Jo !-«-« -f"{(i+«)-'-a+«)-}T-r. Jo fl where y is Eider's constant. 4 Prove that 5. Prove that r («) = Limit n'B («, n). 6. Prove that, when ^ > 1, ^(p, j)+^(p+l, j)+^(p+2, ?)+...-5(p, 9-1). 7. Prove that Biscay q) gjr a(a + l)9(g + l) ^(i>i?) '*"l>+?"^1.2.(p + j)(p+j + l)"^-- 8. Prove that ^tP, q)B(p+q, r)^B(q, r) B(q + r, p). (Euler.) 9. Prove that log r (*)=(!-«) log ir+y(i-«)-i log sin «irH — 2 -^ — sin2n«9r. IT n-i n (Kummer.) 10. Prove that I co8'>+«"*wco8(p-g)w(iM—; — ; Tv-r^rx:; ,-„-, r. (Cauchv.) 11. Prove that ..»i.(«,)-w(^)*/;'-!;=!^az|L'^.. m^-> 12. Prove that 2»(p.i'+«) 2J— V+2(2^1) + 2".M2p+iK2^3) + "r ^ ^^ 13. Prove that r r 14. Prove that (j»+i)"lP 4p0>+l)^2.4».^0j+l)(p+2)^-; • \ V(p) ] 2 t 2(2p+l) 2.4.(2p+3)(2p+6)^-r MISC. EXS.] THE GAMMA-FUNCTION. 197 16. Prove that (Binet.) 16. Prove that where y is Eider's constant. (Legendre.) 17. Prove that B(p,p)B{p+i,p+i)^,^^. (Binet) 18. If shew that / \ogT{z)dz^u, du , and hence (or otherwise) that ti^orlog J?— ^+^log29r. (Raabe.) 19. Prove that, for all values of t except negative real values, logr«-(f-i)log^-«+ilog(2^)+ i rTTl- Sin %nirx (Bourguet.) n^r 20. Prove that ,_ r(a-hl)r(a+6+c4-l) Pfia-f^KLif!)^^ *'^r(a+6+l)r(a+c+l)"jo~;^ ' l ^- (l-^)log- 21. Prove that ri^«-i-^-i ^ V 2 / \2/ Jo(l-H^)loga?''^^ /a\ //3-Hy (Kummer.) 22. Prove that , r(a+6+l)r(a-Hc+l)r(6H-c-H) f Ml -^)a-^)(l-^) ^_ '^r(a+l)r(6+l)r(c+l)r(a+6i-c+l)";o ,, M 1 (l-x)log- 23. When a? is positive, shew that - T{x) • 2nl 1 v;;-^w 2 (Cambridge Mathematical Tripos, Part I, 1897.) 24. If a is positive, shew that r(g)r(a+l) ^ « (~ l)''a (g- 1) (a- 2) ... {a-n) 1 r(«+a) ^ »=o n\ « + »* 198 TRANSCENDENTAL FUNCTIONS. [CHAP. IX. 25. Shew that 26. The curve f^ «= 2** ~^ cosing is composed of m equal closed loops. Shew that the length of the arc of half of one of the loops is 1 f' -i-1 2^"^.—. / (cosarV dx, and hence that the total perimeter of the curve is 27. Prove that logr(«)=(«-i)log«-«+ilog(2fl-) 28. Prove that 29. Prove that 30. Prove that a- a* log r («+a)=log r («)+a log « - 2« a / a(l — a)<ia- / a{\^a)da Jo Jo 2s (2+1) a/ a(l-a)(2-a)flfot- / a(l-a)(2-a)tia 3«(«+l)(«+2) 31. Prove that 32. Prove that 5(p.?)-(^r,!.i(2-)*»*(p.«, MISC. EXS.] where THE OAMMA-FUNCTION. 199 M(p,q)^2pj*^ dt e^i^-l (p+?)r and p*— ^+^+/>S'. 33. Expand {r(a)}-i as a series of ascending powers of a. (Various evaluations of the coefficients in this expansion have heen given by Bourguet, Bull, des Sci. Math, v. (1881), p. 43; Boiu^uet, Acta Math, ii. (1883), p. 261 ; Schlttmilch, Zeit$ohrift fiir Math, xxv. (1880), pp. 35, 851.) 34. Shew that and where r^^-a«oos 6a?<iir-co8 {(m + 1) («■ - <^)} ?^-^^^^ r^^—xsin &p<iir-sin {(m+ 1) (ir - 4)) ^^j- , - a-k-bi^r (cos 0+t sin ff>). 35. If I e-'tf^idty shew that and Pi^)--] ^^^^ X l!a?+l'*"2la?+2 3!^+3"*"*"' P(x+l)-arP(ar)-i. 36. Prove that d i-, r(z+ar) X ,x {x-l) . x{x-'\){x-'%) 5^*"* r(z) "« ^«(«+i)"*"^ «(«+i)(«+2) "^•••* 37. If a is negative, and if where v is an integer and a is positive, shew that r^r) r (a) r( where (-l)»(a-l)(a-2)...(a-n)^ 0.(,).gW-0(-«), (Hermite.) 200 TRANSCENDENTAL FUNCTION& [CHAP. IX. 38. When - oo <a< 1, shew that r{x)T(a-x) % K ; R* r(a) ",!, ^+;i-,*,^::^:rn' ^^ (-l)»*a(aH-l)... (aH-n-1) 39. When a > 1, and y and a are reepectively the integral and fractional parts of a, shew that T{x)T{a-x) ^ J G{x)p^ • Q{x)p^^^ L^— a JP— a— I 4?— a — v+lj where *«-('-!)('-^)-('-,-T^) and _( - l)**o(a+l) ...(a+n-l) 40. If Pi, p^y ... py are the roots of the equation p»' + aip''-i+...+aK«0, shew that • f/ X X* ' ^^ \ *!*) ,r(l)a.Xj^^^^ T (z- pix)r (z- p^) ...r {z- p^)' 41. If a and b are real and positive, prove that / e"i'V-iw«-*-irfttrfv»r(a)r(6). 42. By taking as contour of integration a parabola with its vertex at the origin, derive from the formula ^W = s^ l^^^^dz ^ ' 2tsma9ry the result T{a)'^^^^f e-'^x^'i (I +j^)-^ [3 sin {x+acot'^i-x)} +8in {x+(a - 2) cot-» {'-x)}]dx. (Bourguet.) 43. Prove that, when 1 < « < 2, /*sin6^ - __ ^"* «• and when < « < 1, sm- pcosftg? . _ f-^ Jo A- 2r(«; «■ ) tr«' MISC. EXS.] THE OAMMA-FUNCTION. 201 44. Shew that w 46. If 2« t7» ('-!) and s 2« F-r and if a function F(x) be defined "by the equation shew (1) that F(x) satisfies the equation (2) that for all positive integral values of x, F(x)~T{x\ (3) that F{x) is regular for all finite values of x, (4) that 46. Prove that the function O (x), defined by the equation 6f(a?+l)=(2ir)«e « « n fl+lj e» , has the properties expressed by the equations log ^j^- , — (a» I «-^cotir:J?c2r-a?log2tr, [ (g-l)(a;-2 ) «-l r Q +itn (n+i)-i- (r(«+i)}.- in iij±;ij. (Alexerewsky.) 47. If « is a positive quantity (not necessarily integral), and « is a real quantity between - ^ and ^ , shew that COS* 1 r(«+i) f. « „ . «(#-2) . . \ and draw graphs of the series and of the function cos's. 202 TRANSCENDENTAL FUNCTTIONS. [CHAP. IX. 48. Obtain the expansion and find the values of x for which it is applicable. 49. Prove that (Cauchy.) i.(-5)' '^^" 60. If ('•*^-rV)/o where | ^ | < I and the real part of or is positive, shew that and Limit (1 - x)^-* f («, ^) = r (1 - *). ae-l 61. If Xf Wy and « be real, and < t9 < 1, and « > 1, and if shew that and 00 ^iri* *(«.,;r,l-,).^J ^'' I. (Lerch.) 62. If shew that tf'-l ' (3) r(|) n-iCW-rQ^^ ,r*Tf(l-,). 63. Let the function <f>i''>(x) be defined by the equation (-!)•*«( I n(i-«— ^) where 5 is an integer ^ m, the function x (0 ^^ defined by the equation and the quantities a^ are constants whose real part is positive. MISC. EX&] THE QABfMA-FUNCTION. 203 Shew that ^t*) (x) can be expressed by the series <^«(^)-2/W(jp+M^); where t9«>ZXrart and where (-i)'/«(«)-j'"x(0 «•«-"*• Shew also that ^t*) (x) satisfies the functional equation M.»' Shew further that when ;^(r)«Bl, <f>i')(x) becomes a function ^^(x\ which has the multiplication-theorem where all the quantities X vary from to (n- 1). (Pincherle.) 64. If where /n\ n! W"rl (n-r)!' shew that . riy)r(sf'-x+n)T(x+v)T{v+n) /nwy, «^; r(y-i?)r(y+n)r(t^)r(a?+v+n)' and that r(y)r(a?+v) (y-x-l)r(-n)r(a?+l)r(y+t;+n-l)' (Saalschiits.) CHAPTER X. Lbgendbb Functions. 111. Definition of Legendre polynomials. The expression (1 — 2zh + /i*)"^ can, when | A { is sufficiently small, be expanded by the multinomial theorem as a series of ascending powers of h, in the form where Pi {z) = Zy -P« (^) = ^^-^— ^ . etc. The expressions Pi{z), P,(^) ..., which are clearly all polynomials in z, are known as Legendre polynomials, Pn (z) is called the Legendre polynomial of order n. It will appear later (§ 116) that these polynomials are particular oases of a more extended class of functions, known as Legendre functions. Example 1. Prove that P„(coe^)=^ — r^coseC*-^^^ ,/ ^' " ^ n ! a (cot B)^ (Cambridge Mathematical Tripos, Part II, 1893.) Let ^ be an angle such that Then (1- -2A cos^+A«)" 4_8in ff "~ sin ^ * cot^« -r-=-^- * coe^-^ ~ sin^ ■scot^-A cosec B. Ill, 112] LBGENDBE FUNCTIONa 205 Therefore by Taylor's theorem we have • iK « ( - A cosec ^)» €?* (sin ^) sin ff ^* ;; « V . ^x" • n n\ (f(oot^)»' or /, or ii.nv-* ^(-A)*coseo»+»^flP«(8m^) (l-2Acoe^+A») *=S^-^ir! d^r Equating coefficients of A*, we obtain the required result. Example 2. Shew that For />-©'■ Therefore Thus (l-2A*+A«)-*=ir-* r «-(i-«'+*^««cft=ir-* f * tf-(i-'«)««6-(»-*y««(&. y —00 J —00 whence the result follows. Example 3. By equating coefficients of powers of A in the expansion (l-2Aco8^+A*)* \ * 2.4 / X (n-iAe-«+^A«d-««+...), shew that 112. SchldfiVs integral for Pn{z\ Let h be any quantity which is not greater than the radius of convergence 00 of the series 2 h^Pn {z\ Then (1 — 2zh + A*)~* can be expanded as the series 1+AP^(^) + A'P«(^)+A»P3(^)+.... But (1 - 2zh + A')~* is the residue, at the pole 1 (1 - 2zh + /i')* ^"h" h of the function — 2A""* "^ I ^ "" t J Ti r • 206 TEANSCENDENTAL FUNCTIONS. [CHAP. X. Now the last expression has two poles, namely at the points 1 (1 - 2zh + h*^ and t^Ud^^^^, ft n When h is very small, the former of these poles is close to the point t = z, while the second pole is in the infinitely distant part of the plane. Therefore, if (7 be a contour in the ^-plane, including the point z, the former pole only is contained within C when h is not large, and so we have di. Equating coefficients of A", we have the result which is called Schldflis integral-formula for the Legendre polynomials 113. Rodrigues* formula for the Legendre polynomials. From Schlafli's integral dt we immediately deduce, by the theorem of § 38, the result which is called Rodrigvss formula, 114. Legendre'a differential equxttion. We shall now prove that the function y = Pn (-^) is a solution of the differential equation which is called Legendre's differential equation of order n. * Schlafli, Ueber die beiden Heine*$cheii Kugelfunctionen ; Bern, ISSl. 113—116] LEOENDRE FUNCTIONS. 207 For on substituting Schlafli's integral, we have ds^ dz 27n and this integral is zero, since the function (<* — 1)**"*"^ {t — ^)~"~' resumes its original value after describing the contour (7. The Legendre pol3n[iomial therefore satisfies the differential equation. The differential equation can clearly be written in the alternative form 116. The integral-properties of the Legendre polynomials. We shall now shew that and that j[Pm{z)Pn{z)dz = 0, if m and n are positive integers and m is not equal to n. dPn] For since we have + (m-n)(m + n-l)\ PnPndz^O. ■ —1 Integrating by parts, this equation gives - (m - n) (m + n - !)/'-?». («) -P» («) dz -['a-^){p„(.)^-P„w^)}; which shews that the integral j' P„(z)Pn{z)dz has the value zero when m is not equal to n. = 0, 208 TRANSCENDENTAL FUNCTIONS. [CHAP. X. To establish the second part of the theorem, let the equation be squared, and the resulting equation integrated between the limits — 1 and + 1 ; using the result already proved in the first part of the theorem, we thus obtain or Equating coeflScients of h^ in this equality, we have which is the desired result. Example 1. Prove that, if m is not equal to n, x{l +(-!)*+«}. (Cambridge Mathematical Tripoe, Part I, 1897.) Example 2. Prove that j-i dzr d^ ^^ '^ '^^'''^ (27i + l)(n-r)!' according as m, n are unequal or equal. (Cambridge Mathematical Tripos, Part I, 1893.) 116. Legendre functions. Hitherto we have supposed that the index n of Pn{z) is a positive integer; in fact, Pn{z) has not been defined except when n is a positive integer. We shall now see how the definition can be extended so as to furnish a definition of Pn{^)» even when n is not integral. An analogy can be drawn from the theory of the Gamma-function. The expression zl as ordinarily defined (viz. as « («— 1) (*-2)...2. 1) has a meaning only for positive integral values of z; but when the Gamma-function has been introduced, z\ can be defined to be r (2 + 1)) and so a function z\ will exist for all values of z. Referring to § 114, we see the differential equation 116] LEQENDRS FUNCTIONS. 209 is satisfied by the expression • even when n is not a positive integer, provided that (7 is a contour such that the function (^ - l)n-fi {t - xr)»+a resumes its original value after describing C. Suppose then that n is no longer taken to be a positive integer. Now the function (f*— 1)*+* (t — xr)-**-« has three singularities, namely the points t^l,t^^l,t»z; and it is clear that after describing a small closed contour enclosing the point ^^l, the function resumes its original value multiplied by e»»»<»+i) ; while after describing a small closed contour enclosing the point t = z, the function resumes its original value multiplied by If therefore (7 be a simple contour enclosing the points t^ I and t ^ z, but not enclosing the point ^=: — 1, then the function (e» - l)n+i Ji+a (t-z) will after describing C resume its original value multiplied by e"***, i.e. it will resume its original value. Hence whatever n he, Hie Legendrian differential equation of order w, is satisfied hy the expression where is a simple contour in the t-plane enclosing the points t^l and t^g, hut not enclosing the point ^ » — 1. This expression will he denoted hy P» {z\ and will he termed the Legendre function of the first kind and of order n. We have thus obtained a definition of Pn {z) which is valid even when n is not integral. The Legendre function is a mere polynomial when n is integral, but is a new transcendental function when w is not integral ; just as F {z) is the polynomial (^-* 1)1 when z is integral, but is a transcendental function when z is not integral. W. A. 1* 210 TRANSCENDENTAL FUNCTIONS. [CHAP. X. We shall suppose the many-valued function ^, which occurs in the defining integral, to have the value 1 when z is equal to 1, and when z is not equal to 1 to have that value which would be obtained by con- tinuation along a rectilinear path from the point 1 to the point z, 117. The Recurrence-formulae, We proceed to establish a group of formulae which connect Legendre functions of different orders. We have by § 116, for all real or complex values of w, m Integrating by parts, we have 27ri;e2»-(<-«)" and hence we have f „(.)-.P^. (^)- 2^/, 2^^(7-1;^. ^^ <^)- Differentiating this equality, we obtain —dz '~~d'z ■^"-' ^'^ -~2^lc S^'it-zr*^ - ^" ^^^»-' ^^^' 80 ^>-.%i^>-nP„_.(.) (I). This is the first of the required formulae. Next, from the identity we deduce or r (f'-l)»-> ^,, f 2 Kf - 1) + 1} (n - 1) (f - If-' dt [ (n - 1) {(t -z) + z] (f - ly-' ^, 117] LEGENOEE FUNCTIONS. 211 or or (n-i)^r o*-iy'-^ 27n; ./c2'*-H^-'8^V or, by formula (A) above and Schlafli's formula, = n{PnW--^Pn-iW}+(w-l)Pn-.(^)-(^-l)^Pn-iW, or nPnW-(2n-l)5Pn-i(^) + (n-l)Pn^(^) = (H), a relation connecting three Legendre functions of consecutive orders. This is the second of the required formulae. Other formulae can be deduced from (I) and (II) in the following way : Differentiating (II), we have "" --^— -(2n^l)^ d^ "^^^"^^"d^ =(2n-l)P^,(4 Substituting for dPn (Z) dz from (I), we have f dPn-x^) p ,1 dPn-x(^)^,^ ,, dPn->(^) = (2n-l)P„_,(^), or -(n-l)r^^g<l> + (»-l)^^g^^ = -(n-l)'P^,(4 or ^ —Tz -^^ =(n-l)P„-,(4 Changing (n — 1) to n in this equality, we have Next, changing n to (n + 1) in (I), we have 14—2 r^^^BiB^BR 212 TRANSCBNDENTAL FUNCTIONS. [CHAP. X. Adding this to (III), we have ^-P^^(^) - ^-P^^(^) ,(2n + l)Pn(^) (IV). Lastly, combining (I) and (III), we obtain the result {ii>-l)^^^nzPniz)-nP^r{z) (V). The formulae (I) to (V) are called the recurrence'/ormtdde. The above proof holds whether n is an integer or not, Le. it is applicable to the general Legendre functions. Another proof which, however, only applies to the case when n is a positive integer (i.e. is only applicable to the L^;endre polynomials) is as follows : Write Then equating coefficients of powers of A in the equality (l-2Az+A«)|^=(«-A)r, we have nP,W-(2n-l)«P^_iW+(n-l)P,,.,W-0, which is the formula (II). Similarly, equating coefficients of powers of A in the equality we have which is the formula (III). The others can be deduced from these. Eixtmple, Shew that, for all values of n, (2n+3)P«,^i-(2n+l)P.«=^{z(P,«+P«,+i)-2P»A+J. (Hargreaves.) For ^{^W+i^n*i)-2P,P,^J =P.«+P«..,+2.P.5^+2.P..,%^-2P.%.2P.,,f^ =P,«+i»,+,+2P,(-n-l)i>.+2P,^i(n+l)i'.+ , (as is seen by using formulae (I) and (III)) -(2»+3)i«,^,-(2n+l)P,«, which is the required result 118] LEGENDRE FUNCTIONS. 218 118. EvaiuaUon of the integral-expression for Pn (z), as a power-series. When n is a positive integer, we have seen that Pn {z) is a polynomial in z. When n is not a positive integer, however, P^ {z) is not a polynomial ; and as P^ (z) is not a regular function of z for all finite values of z unless n is integral, it follows that no power-series exists which represents P^ (z) for all finite values of z, when n is not integral. In order to find a power-series capable of representing Pn(z), we must therefore make some supposition regarding the part of the 5-plane on which the point z lies. We shall suppose that z lies within a circle of radius 2, whose centre is the point 1 ; so that |1-^|<2. As the contour C of § 116 was subject ooly to the condition of enclosiug ^ a> 1 and t^z without enclosing t » - 1, it is clear that we can choose it so as to lie entirely within the circle of centre 1 and radius 2 in the f-plane, ie. to be such that the inequality 1 1 * f | < 2 is satisfied for all points t on C. Now write ^— 1 = (^ — l)t^. When t describes the contour C, the point representing the variable u will describe a contour 7 on the t£-plane ; since C encloses the points t^z and ^ » 1, 7 will enclose the points u^l and u^O; 2 and since |1 — <| < 2, we shall have \u\< . ^ . for all points u on 7. Then changing the variable of integration from t to u ih the integral which represents Pn (z), we have ^(^•^V-T^'*^--^)-^^ 2wi.y 2 Since \u\ < -, 77 we can expand this in the form du. Now on integrating by parts, we have the result J y Ly W J Tt J y The first expression on the right-hand side is zero, and so we have I !*•'+*• (u-l)-*^»dw«^^^[ w«'+»-»(w-l)-~-*(u-l)dw n Jy ^ ' n Jy or I w'+« (u - !)-*-> du = ^^i^ I w'"+«-^ {u - 1)-«-* du. 214 TRANSCENDENTAL FUNCTIONS. [CHAP. X. Therefore Now transform the integral on the right-hand side, by writing u ss v-l* r f t?*dt; The integral I w*(tt — l)~*~*(iu becomes — I — -r , where the integration has now to be taken in the positive sense round a contour 8 enclosing the points t; » and t; «= oo , but not enclosing the point v^l. This integral can be replaced by + i --— j , where the integration has to be taken in the positive sense round a contour S^ enclosing the point t;= 1, but not enclosing the points v^O, or t; ae oo (since the integrand has no singularities in the region between the contours B and S^). The contour B' can now be diminished until it becomes an infinitesimal circle surrounding the point t; = 1. The value of the integral is then 1* I -— ^ , where the integration is taken round this contour; or 2m, since the many- valued function v* has been taken to have the meaning 1 at the point t; = 1. We thus have r + n r — 1 + n 1 + n 2mjy ^ -/ -- ^ • ^.1 ••• 1 ' and on substituting this in the expression already found for Pn(^)i we obtain p , X 1 _^ S n(n-l)...(n-r-H) (r-hn)(r-l+n)...(l-f n) fz-lV an expansion of Pn (z) as a series of powers of ('2^—1). If now, as in § 14, a series of the form ^^l.c^^ 1.2c(c-hl) ^+- (a hypergeometric series) be denoted by F(a, 6, c, z\ then the expansion can be written in the form Pn(r)-J'(-n, n + 1, 1, ^^). This is the required expression for P^ (z) as an infinite series. It is valid at all points z within the circle whose equation is |1 — ir| < 2. 119] LEGENBBE FUNCnON& 215 Corollary, Since this series is clearly unaffected when n is changed to — n — 1, we have Note. When n is a positive integer, the above series terminates and gives the expression 1 — « of P» («) as a polynomial in —^ . 119. Laplace's integral-expressuyn for P^ {z). We shall next shew that, for all values of n and for certain v^alues of z, the Legendre function Pn{z) can be represented by the integral (called Laplace's integral), - ['{^ + cos <^ (^ - l^}«d<^. When n is not an integer it is necessary to state which of the branches of the many-valued function in the integrand is to be taken: we shall take that branch of the function [z + cos <f){^ — 1)*}** which reduces to unity when taken by the process of continuation along a straight path to the point z^l. It will appear later that it is immaterial which branch of the two-valued function (^ — 1)* is taken. (A) Proof applicable only to ike Legendre polynomials. When n is a positive integer, the result can easily be obtained in the following way. We have 2 A«P«(^)-(1-2A^H-A«)-*. But (i-2A.+A«)-*=ir^^f-^^ d<f> (l-A^)-A(-2»-l)*cos<^' as is seen by applying the ordinary formula for the integration of d<t> : a + 5cos<^* Expanding the integrand of the integral in ascending powers of h, we have (1 - 2A^ + A«)-* « - i A« ['{z + cos ^ (^«- 1)*}**^^, ^»=o Jo and on equating coefficients of h^ on the two sides of this equation, the required result is obtained. As however the theorem is true whether n is an integer or not (ie. as it is equally true for the Legendre functions and the Legendre polynomials), it is necessary to have a general proof independent of the character of n ; this will now be given. (B) General proof First, we shall shew that Laplace's integral satisfies Legendre's equation. I^^p*^^^^^^" 216 TRANSCENDENTAL FUNCHONa [CHAP. X. For if we write 'fJo we have = - ['{^H- COS <^(^«- !)*}«-» {n8in»^-l-^C08^(^-l)-*}d^. But f '{-? + COS <^ (z« - !)*}«-« sin«<^d^ « — {^ + cos ^ (j8* - 1)*}*^ 8in<^ COS ^ 4- Tcos A :^ [sin <^ [-? + cos <^ (i^ -!)*}«-•] d^ « / ' {^ + cos ^ (^ - 1 )*} »-« COS" <^ rf^ - (n - 2) f '{^ + COS <^ (z« - 1)*}'^» cos <^ (-?» - 1)* sia«^d<^ -'o «r{^ + cos^(^«-l)*}~-^ci<^~(n-l)r{xr + cos<^(-8«--l)*}»-^sin«<^d^ Jo Jo + (n - 2) ^ rsm*^dif> {2: + cos ^ (-e* - 1)*}**"*. Jo Therefore n 1 {-ar + cos ^ (-«■ - 1)*}*^ sin«^(i<^ Jo =rf'{2r + cos^(-e»-l)*}'«d<^ + (n--2)2rr{ir + cos<^(-^«-l)*}~-«sin«<^^^ Jo Jo . Thus we have (l-^)g-2.| + n(„ + l)y = - (n - 2) « f ' {« + COB (^ («• - !)*}"-• sin'^d^ 9r Jo - - « (^ - 1)-* f'fz + cos ^(«' - l)»}»-'cos <tKi4> = « !L ^ (^ _ 1)-* f ^ [U + cos ^ (^ - 1)*}"-* sin ^] # TT Jo w<P -0, 119] LEGENDRE FUNCTIONS. 217 which shews that Laplace's integral satisfies Legendre's equation, whatever n and z may be. 1 -^z Now suppose that z is nearly unity, and put — ^— » u. Then the integral becomes 1 r* -I {l-2w + cos^(-4w4-ti«)*J«d<^, which for small values of u can be expanded in the form 1+- d<i 2 -^^ ^- — j^^ ^ {-2a + co8 A (-4w + w*)*K This is a series of powers of m*; the first terms (neglecting w*) are If* If' 1 + 2inw* - I cos ^d,^ — %nu - / {1 + (n — 1) cos'^} d^, or 1 — inu ^ , 2 or 1 — w (w 4- 1) «. It is clear that odd powers of 'v^ can arise only in conjunction with odd powers of cos^ in the integrand, and so here vanish when integrated. Laplace's integral can therefore, when u is small, be expanded in ascending powers of u in the form 1 — n (n + 1) w + OjW* + OjW* + a4W* + ... . But the coefficients a,, a,, ... can be found by substituting this expression in Legendre's equation, and equating to zero the coefficients of each power of u. We thus find that / txy ^(^-~l)»"(^^y + 1)'(1 -l-n)...(r — 1 +n)(r-hn) r I r : and thus Laplace's integral is equal to jP(-n, n + l, 1, -^)» or (§118) to .PnW. We thus have, for all real or complex values of n, the result T^(z) « 1 ['{^ + cos <^ {z^ - 1)*}» d^. It must be observed that as the power-series Fi-^n^ « + l, 1, — g— j was used in the proof, this proof is valid only for values of z which satisfy 218 TRANSCENDENTAL PUNCTrONS. [CHAP. X. 1 1 — i^l the inequality - — ^ — < !• ^^ however Pn (z) is an analytic function of z, 'the result will be true for a more extended region including this, provided the integral is an analytic function of z within this more extended region: since if these two expressions are equal for any region however small in which they are analytic functions, they must be always equal so long as they remain analytic functions. But it is easily seen that for the integral i J'{^ + cos <^ (^« - l)*j« d<^, every point on the imaginary axis in the -^-plane is a singularity: and therefore the region in the ^^-plane for which the equality "Jo is established is the region for which the real part of z is positive. Corollary. Since we have for all values of n, real or complex, the result Pn (Z) = - ['[Z + COS <^ (^ - l)»}-^» d<l>, TrjQ so long as the real part of ^ is positive. Example, If shew that (Binet) 120. Tlie Mehler-Dirichlet definite integral for P« (z). Another expression for the Legendre function as a definite integral may be obtained in the following way. For all values of », we have by the preceding theorem P„ (^) = 1 Tf^r + cos ^ (^ -!)*}« d<^ In this integral, replace the variable <^ by a new variable h, defined by the equation A «= ^ + (^ — 1)*C08 ij> 120] LEGENDRE FUNCTIONS. 219 80 that and We thus have ctt«-(^-l)i8in<^d^, i(l-2A«: + A«)*-:(^-l)*8in<^. and therefore (1-2A^ + A»)* d<f>, Now write z = cos 0. Thus Pn(co8^)«- C^A'*(l-2Acos^ + A«)-*dA. Writing A « «•♦, this becomes ^^^'^^^~^j.,(2cos<^-2cos^)*' or i> / ^\ 2 r^ cos (^ + i) i , , Pn (cos ^) » - I 7^7 ^T ^loi #. ^ ' ^ J {2 (cos <^ — cos 0)}^ ^ This is known as Mehlers simplified form of Dirichlet's integral. The result is valid for all values of n. Example 1. Prove that, when n is a positive integer, *^ ' irj^ {2(cosd-co8<^)}* For we have Put r* dw IT Jo a+6-(a-6)co8ir*2a^*' a-(H-A)«, 6=l-2Ay+A«, The equation becomes IT /-y (i±.^)f^ Writing £«cos <^, y>«coB ^, this gives (l-2Acos^+A<)-*=- f'(l+A)8in<^(l-2Aco8<^+A2)(l+co8<^)-i(oo8^-co8<^)-*ci0. ir J Equating coefficients of A* on both sides, we have P,(cos^=^| ^«in(n+^)<^sin<^ {2 (cos ^- cos 0)}* ^ ]. r* sin (nH I sm^coS"^ ./ A 2 2 220 TRilNSCENDENTAL FUNCTIONS. [chap. X. or P»(co8^-- I {2 (006^-008 0)}-4 sin (n+i)<^. Example 2. Prove that PAoo.e).±.f^ 2Aco8^+l)i dh. the integral being taken along a closed path which encloses the two points h^e*^^ and the conventional meaning being assigned to the radical. Hence (or otherwise) prove that, if ^ lie between }ir and }ir, 2.4... 2n ^-^^^^^-nzr^ik^) coa(nB+<t>) V C06(n^-h3<^) (2sintf)* ''■2(2«+3)~'(2sin^)l 1«.3* oos(n^+6<^) where <t> denotes i^-^ir. Shew also that the first few terms of the series give an approximate value of F^{cob$) for all values of $ between and n which are not nearly equal to either or n. And explain how this theorem may be used to approximate to the roots of the equation P^ (cos ^=0. (Cambridge Mathematical Tripos, Part II, 1895.) 121. Expansion of Pn(z) as a aeries of powers of - , z We now proceed to find an expansion of the Legendre function which is valid for large values of z. If the real part of 2^ be positive, we have for all values of n (fix>m Laplace's integral) ^ Jo Now suppose that |-?| is very large : then this can be written in the form Expanding the integrand in ascending powers of - , this gives z We can evaluate / (1 + cos <^)'* d<^ and | cos<^(l + co8<^)'*-*(i</> ^0 Jo 121, 122] LEGENDBE FUNCTION& 221 by patting ^ » 2*^ and using the result and thus we find that P» {t) can be expressed by a series of powers of - , the first two terms of the expansion being given by the equation P .V 2»5"r(n+^) ( _ n(n-l) ) The general law of the coefficients in the series can without difficulty be found by substituting in Legendre's differential equation (§ 114) ; and iu this way we find that P» (z) can be eaypressed by the hypergeometric series P/^^ = 2VM[>^) /1-n n 1 1\ in the notation of § 14. This series has only been proved to hold when z is large and the real part of <g: is positive : but by § 14 it converges, and so represents an analytical function, over all the area outside the circle of centre and radius 1. The series therefore represents Pn(z) over this region. 122. The Legendre functions of the second kind. Hitherto we have considered only one solution of the Legendre differential equation, namely Pn (z). We can now proceed to find a second solution. It appears from § 114, that the differential equation is satisfied by the integral ^t* - l)"" (t - zy-' dt, /< taken round any contour such that the integrand resumes its initial value after making the circuit of it. Let D be a figure-of-eight contour in the ^plane, enclosing the point t^ + 1 in one loop and the point t = — 1 in the other, and not enclosing the point t^z. Then after describing this contour, the above integrand clearly resumes its initial value, since it acquires the £su;tor e^ after describing the first loop, and this is destroyed by the &ctor e"**** acquired during the description of the second loop. D is therefore a possible contour. A 222 TRANSCENDENTAL FUNCTIONS. [CHAP. X. A solution of Legendre's equation is therefore furnished by the function Qn {z)y if Qn {z) be defined by the equation it is supposed that, in describing D, the point t makes a positive, i.e. counter- clockwise, turn round the point ^ = — 1, and then a negative, i.e. clockwise, turn round the point ^ » + 1. The significance of the many- valued functions (^• — 1)" and {t^z)"^"^ will be supposed to be fixed in the same way as before. Another form of the integral may be obtained in the following way. Let the contour become so attenuated as to consist simply of a line joining the points — 1 and + 1, described twice, and two small circles round the points — 1 and + 1 : when the real part of (n -h 1) is positive, the parts of the integ^l arising from these two loops are at once seen to be infinitesi- mal; and thus we have = 2isin?i7rJ {l-fYit-zY^'^dt, «o Qn{z)^^,\\\-i^T{z-t)^'di, This last result is valid when the real part of (w + 1) is positive. When n is a positive integer, the original definition of Qn{z) becomes imdeterminate: in this case we can use the formulae just found. Qn{z) is called the Legendre function of the second kind and of order n. 123. Expansion of Qn (z) as a power-series. We now proceed to express the Legendre function of the second kind as . . 1 a power-senes m - . We have, when the real part of (71 -f- 1) is positive. Suppose that 1 ^ 1 > 1. Then the integral can be expanded in the form «.(«)-ss^ /la -«■)■ (1 -;)""' <» 123] LEO£NDBE FUNCTIONS. 223 as is seen on writing r for 28, since the integrals arising from odd values of r obviously vanish. Writing ^ « M, we can evaluate the coeflScients of powers of - as follows : z r(n + i)r_(* +i) r(n + « + f) "' and thus the formula for Qn {z) becomes Q (^\J'^^ r(n4-l) 1 y/n-hl 71 + 2 3 1\ VnW 2»^ir(n + f)^«+i^ V 2 ' 2 ' ^^r z'J' This is the expansion of the Legendre function of the second kind as a power-series in - , corresponding to the expansion obtained for Pn (-?) in § 121. z The proof given above applies only when the real part of (n + 1) is positive ; but a similar process can be applied to the integral ^ ^ 4i sin riTT j j[) 2^ ^ / \ / the coefficients being evaluated in the same way as those which occurred in the expansion of the Legendre function Pn (^) in ascending powers of —5— ; the same result is reached, which shews that the formula O r^^ '^^ r(n-H) 1 r^/n-H n + 2 3 1\ VfiW*2«+ir(n + f);?«+^^ V 2 ' 2 ' ^"^2' W is true for all values of n, real or complex, and for all values of z represented by points outside the circle of centre and radius unity. Example 1. Shew that, when n is a positive integer, We can write Legendre's differential equation in the form (l-^g-2*|+n(«+l)«-0. It is easily verified that this equation can be derived from the equation (!-«») 5+2(»-l)«J+2«*=0, by differentiating n times and writing t£» ;^ • 224 TRANSCENDENTAL FUNCTIONS. [CHAP. X* Now one solution of the latter equation is a7«(2'- 1)* ; and a second solution oan be derived bj the ordinary process for finding a second solution of a linear differential equation of the second order, of which one solution is known. Thus two independent solutions of this equation are found to be It follows that (««-l)» and (««-l)» I (t^-l)-«-irfr. is a solution of L^;endre's equation. As this expression, when expanded in ascending powers of - , commences with a term in d'^'^ it must be a constant multiple of Q^ (t) ; and on comparing the coefficient of «"*~* in tnis expression with the coefficient of z"^"^ in the expansion of Q^ (')» ^ found above, we obtain the required result. Example 2. Shew that, when n is a positive integer, the Legendre function of the second kind can be expressed bj the formula For on expanding the int^;rand (v^-l)"'"'^ in ascending powers of -, the right-hand V side of the equation takes the form ^" ax- /.■'*>"■ {^' - SSI- '^^^m^ -..} . and on performing the integrations this becomes n! f 1 ■ (n-H)(7t+2) 1 1 , (2» + l)(2»-l)...3.1 |«* + i"*" 2(2»+3) z^+s -«■—/» or §«(«). Example 3. Shew that, when n is a positive integer. This result can be obtained by applying the general integration-theorem to the preceding result. 124. The recwrrence-formvlae for the Legendre function of the second kind. The functions PnW and Qn(^) bave been defined by integrals of pre- cisely the same form, namely /(t«-i)»(^-^)-«-id^. It follows therefore that the general proof of the recurrence-formulae for Pn(^)t given in § 117, is equally applicable to the function Qn(^)'i a^id hence that the Legendre function of the second kind satisfies the recurrence-fbrmtdae 124, 125] LEGENDRE FUNCTIONS. 22& dQn(z) dQ^Az) ^Z g^' = n(2^.(4 , dz nQn (z) - (2w - 1 ) zQ^, (z) + (n-l) Q^ (z) = 0, , dQn (Z) dQn-r (Z) _ ^^ , . dQn+1 JZ) dQn-i (Z) ,^_ . , , ^ . -rfi d^^ (2n+l)Q„(.). ft (^ - 1) ^^ = nzQ„ (z) - riQ^, (z). 126. Laplace's integral for the Legendre function of the second kind. Consider the expression y = f {z + cosh (z* - 1)*}-**-^ d0, Jo in which z is supposed not to be a real negative number between — 1 and — 00, and the real part of (n + l) is supposed to be positive; under these conditions the integral certainly exists. If now we form the quantity (which occurs in Legendre's diflferential equation), we find for it the value - (n + ly i {z-k- (f - 1)* cosh ^}-^« sinh* Odd Jo + (w + 1) f {2: + («» - 1)* cosh e]-"*-* dd Jo 4- (n + 1) 2r («« - 1)-* f {^ + (^ - 1)* cosh ^j-"-' cosh dd0. Jo This expression can be transformed, by integration by parts, in exactly the same manner as the corresponding expression found in the discussion of Laplace's integral for Pn (z), in § 1 19 ; and thus it is found to be zero. The quantity y therefore satisfies Legendre's equation. In order to compare y with the solutions Pn (z) and Q» (z) which have already been found, we suppose that | ^ | is large, and write y in the form g-nr-i J*|l+cosh^(l-^ + ...)p 'd0, W. A. 15 226 TRANSCENDENTAL FUNCTIONS. [CHAP. X. which when expanded as a power-series becomes ^0 a^ a^ ch^ m where ao= f (1 + cosh ^)-«-i d^ ^0 /, '^,.v'(l-v)-*dv. where . = j-j-^. = ^— rr-8(^ + l> i)> where B is the Eulerian 2n+i integral of the first kind, Now any expression of the form (1) which satisfies Legendre's differential equation must be a multiple of On(^) (since, by substituting the expansion in the differential equation, we can determine the coefficients ai,a^,a^,.,, uniquely in terms of a©, which shews that all expressions of the kind are multiples of any one of them); and as the value found for ao is equal to the coefficient of the initial term in the expansion of Qn (z\ we have y^Qnizy Thus we have the result Qn W = [ {^ + (^* - 1)* cosh ^}-*-i d0, which may be regarded as the analogue of the Laplace's integral already found (§119) for Pn(^). The theorem is valid only when the real part of (n + 1) is positive ; and the proof has assumed that | ar | > 1 ; but the equivalence of Q„ (z) and the integral, having been proved to subsist for this range of values of z, must continue to subsist for all values of z, continuous with this range, for which the integral continues to represent an analytic function of z ; and hence the theorem holds for all values of z except those which are real and less than — 1, which are singularities of the integral 126. Relation between Pn(z) and Qn(z), when n is integral. When n is a positive integer, and z is not a real number between 1 and — 1, the functions Qn{z) and Pn(z) are connected by the relation which we shall now establish. 126] LEGENDRE FUNCTIONS. 227 When 1 ^ 1 > 1, we have l/>«.4^-U>w^('-!-^^••■)• Now if (n + A) is an odd integer, we have r -P" (y) ^^y = f '-P- (y) y^ - ?p- <y) ^^y = «. J -1 Jo Jo If n is less than ifc, and (n 4- A;) is an even integer, we have \ \\Pn (y) y'dy = l^n (y) y'dy 1 fi d^ (by Rodrigues' theorem) = g^j J^ y* ^^n (y' - 1)" dy 1 f^ (integrating by parts) = ^^ A; (A; - 1) . . . (Ar - w +1) y*"* (1 - ff dy JL Til J =2^!^'^*""^^^*""^)-(*-^-^^^^(^r^' ^ + i) - A?(A;-1)(Aj-2)... (Jk-n + 1) '"(fc + n+ l)(A; + ?i-l)...(A?-n+l)' If on the other hand h is less than n, and (n H- &) is an even integer, the same process shews that the integral vanishes. Therefore A;(A;-l)...(i-n4-l) 1 2J.i '^^^^i^-y"" (A: + n4- 1) (i 4. n - 1)... (A;-n+l) -e*+i ' where the summation is taken for the values A; = n, n 4- 2, 71 + 4, n + 6, ... 00 . But this expansion, by § 123, represents Q,n{s^- The theorem is thus established for the case in which |£:| > 1. Since each side of the equation Qn(.) = |/;P„(y);^ represents an analytic function even when \z\ is not greater than unity, provided z is not a real number between — 1 and 4- 1, it follows that, with this exception, the result is true universally. Example, Shew that Q« {z\ where n is a positive integer, is the coefficient of A* in the expansion of(l-2A«+A*)-ico8h-i j- ^ " \ . For n=o 11=0 * 7 -I *— y _1 n (l-2Ay+A«)~*d y =(1 -Steft+A«)-*co8h-> {-f J J J • 16—2 228 TRANSCENDENTAL FUNCTIONS. [CHAP. X. 127. Development of the function (t^x)"^ as a series of Legendre polynomials in x. We shall now obtain an expansion which will serve as the basis of a general class of expansions involving Legendre functions. We have, by the recurrence-formulae, (2n + l)a:Pn(a:)-(n+l)Pn+i(a?)-7iP,^i(a:) = 0, (2n + 1) zPn {z) - (n 4- 1) Pn+i {z) - nP^i {z) - 0. Multiply the firat of these equations by Pn{^\ the second by Pn(^)> and subtract: we thus obtain (2n + l){z^x)Pn{z)Pn{x) = (n + 1) {Pn^, (Z) Pn {X) - Pn (z) Pn^i (x)} - n {Pn (Z) Pr^, (X) « Pn (x) P^^ (z)]. Write n = 0, 1, 2, 3, ... n successively, and add the resulting equations. This gives {P,(x)P,(z)-^SP^(x)P,(z)+...-\-(2n-\^l)Pn{x)Pn(z)}(z-x) = (n + 1) {Pn+i (Z) Pn (^) - Pn+1 (^) Pn (z)]. Divide throughout by (z — x){z — 1\ and integrate from -? = — 1 to ^ = + 1. Thus x\'^\2r^\)Pr{x)^^dz ^r., {.z-x)iz- i) ^^»^- <^> ^» ^"'^ ■ ^•^' ^''^ ^" ^^>' "^ (by partial fractions) = — — I -— — {Pn+i (-8^) Pn (^) - Pn+i (^) Pn (^)} d-^ "/I't^ {^»+i (^) ^» (^) - -Pn+i («') i^n (^)l d^ . Now by the result of the last article, the left-hand side of this equation can be written -2l(2r-hl)P,(a;)e,(0. In the first integral on the right-hand side, replace the integrand by its n value 2(2r H- l)Pr(a?) Pr{z), and integrate : only the first term survives, since I Pr{z)dz^O, when r is an integer greater than zero ; so the integral has the value 2. 127] LEGENDRE FUNCTIONS. 229 We thus have 2 (2r + 1) P, (w) Qr (t) = -L. + }±l {P„ (x) Qn+, (t) -P„+. (x) Q„ (0). This equation is valid for all values of n. Let us now see if x and t can be so chosen as to make the last part of the right-hand side tend to zero as n tends to infinity. We have, from Laplace's formulae for the functions Pn and Qn, p. w «.„ «) - p... (.) «. «) -If J] {i^i^r ^ #"*. where A denotes a quantity which is finite and independent of n. It is clear that this double-integral tends to zero only when, for all values of (f> between zero and tt, and all values of -^ between zero and infinity, the inequality a;-|-(a^«--l)*cos<^ t-h(e*-l)*coshi|r <1 ^ = K^^i t 'Ih^i)' is satisfied. Writing the inequality becomes uA viu — I cos<f> u \ uj ^ The left-hand side of this relation has its maximum value when cos^s 1, the value being 2 1 u |. The right-hand side similarly has a minimum value equal to 2 { t; {. The condition thus becomes v^ h ft;r--jcosh'^ u\ < \v or |a:H-(^-l)*|<|t-h(<«-l)*|. This inequality shews that the point x must be in the interior of an ellipse, which passes through the point t, and which has the points H-l, — 1 for its foci : for if a be the major axis of this ellipse, then t = a cos + i (a* — 1)* sin 0, where is the eccentric angle of t in the ellipse ; and thus (t^ -. l)i = (a»-- 1)* cos ^-huisin 0, and e + (^» - 1 )* = {a + (a« - 1)*} e^, so that |^ + (^2-i)*| = a + (a»-l)*, and hence the above inequality shews that the semi-axis of the ellipse which passes through x is less than the semi-axis of the ellipse which passes through t, i.e. that x is within the ellipse which passes through t. 230 TRANSCENDENTAL FUNCTIONS. [CHAP. X. Hence if the point x is in the interior of the ellipse which passes through the point t amd has the points + 1,-1, for its foci, then the expansion -i- = I (2« + l)P„(a:)<2„(0 t — X n=0 is valid, 128. Neumann's theorem on the expansion of an arbitrary function in a series of Legendre polynomials. We proceed now to discuss the expansion of any arbitrarily given function in terms of the polynomials of Legendre. The expansion is of special interest, inasmuch as it represents the case which stands next in simplicity to Taylor's series, among expansions in series of polynomiala Let f{z) be any function, which is regular at all points in the interior of an ellipse C, whose foci are at the points -^ = — 1 and z^ + l. We shall shew that it is possible to expand f{z) in a series of the form aoPo(^) + a^Pi {z) 4- a^P^{z) + a^Pj,{z) + ..., where Oq, a,, a,... are independent of z : and that this expansion is valid for all points z in the interior of the ellipse G, For let £r = ^ be any point on the circumference of the ellipse. Then we have or /(^)= S anPn{z\ where an= gTri j /(OQn(0^- This is the required expansion. Another form for a^ can be obtained in the following way. We have _ 2n + 1 rV/<^>^^--2/>^<^>.-^ 2n + 1 /•+! 2 2n + 1 r+i r>'<^^^^\{-Bf f_f{y)Pn{y)dy. 2 The latter is the more usual form for On- 128, 129] LBGENDRE FUNCTIONS. 231 Example 1. Shew that the semi-axes of the ellipse, within which the series converges, are K''+D*"'*K''-J)' where p is the radius of convergence of the series Example 2. If Vy+l/ ' (^+l)(y-l) prove that 129. The associated functions Pf!^{z) and Qn^iz)- We shall now introduce a more extended class of Legendre functions. If m be any positive integer, the quantities will be called the associated Legendre functions X>f the nth degree amd mth order, and will be denoted by Pn^(z) and Qn^(z) respectively. We shall first shew that the associated Legendre functions satisfy a differential equation analogotcs to the Legendre differential equation. For let the Legendre differential equation d^v be differentiated m times, and let v be written for -r-^ . We thus have for v the equation (l--?»)^,-2ir(m + l)^ + (n-m)(nH-m + l)t; = 0. m Write w = v(l-2:*)2; the equation becomes .- -. dhv e, dw ( , , . mM ^ This is the differential equation satisfied by the functions P„«»(^) and Q„'»(4 232 TRANSCENDENTAL FUNCTIONS. [CHAP. X. Several expressions for the associated Legendre functions can be obtained easily from the above definitions. Thus from Schlafli's formula, we have m iiTr* Z J (J where is a simple contour enclosing the points ^^1 and t^z, but not enclosing the point ^ = — 1. From this result, or from Rodrigues' formula, we have, when n is a positive integer, 130. The definite integrals of the associated Legendre fimctions. The theorems already given in § 115, relating to the definite integrals of the Legendre functions, can be generalised so as to be stated in the following form : When m and n are positive integers, j Pn'^{z)P/^(z)dz^O, when r<n, and r^P^''('^}'^'-A,?^r J-i^ ^ 2n + l(n— m)I To establish these results, we use the identity which gives L r (^ + >")' / nm {^(^'-i)*H^<^- '>"}''' (integrating by parts) = aS^^^J' jd^(^-l)f <i' 2 (n + m)! 2n+ 1 {n''m)V i 130, 131] LEGENDRE FUNCTIONS. 233 We can prove in the same way the other result stated, namely that I P„"* (z) Pr"" (js) dz = 0, when r + n. For this integral in the same manner reduces to a multiple of which is zero when n and r are diflFerent. 131. Expression of Pn^{z) as a definite integral of Laplace* s type. The associated Legendre functions can be expressed by means of definite integrals of the same type as those found in § 119 and § 125, as will appear irom the following investigation. We have / {z + cos (^ - !)*}«-*" sin^ (f>d<l> Jo = - {2r4-cos0(^>-l)*}**-^sin*^-^<^cos<^ + 1 COS (f>-j-r [sin««-i (f>{z + cos <^ {z^ - 1)*}**-*^] d<f> Jo a<f> = (2m - 1) f 'cos« <f> sin«^-^ <l>{z+ cos (z^ - l)*}**"^ d(f> Jo - (n - m) ('{z + cos (z^ - l)*}*-«-i cos <^ (z^ - 1)* sin«~ <l>d<f> Jo = (2m - 1) f 'sin**^ (f>{z-\- cos ^ (z^ - l)l}»-«» d<^ - (2771 -1)1' sin»^ { J + cos <^ (^ - 1 )*}«-^ d^ - (71 - m) / (^ H- cos <^ (z^ - 1)*}*-^ sin«'~ <^d0 + (n-m)z I f^r + cos {z^ - l)*}»-«-i 8in»» ^d^. Jo We thus have (n + m) j {z + cos <^ (^ - l)i}»-»» sin*» <^d<^ = (2m - 1) Tsin^^-^ <f>{z + cos <^ (-g« - 1)*}*»-^ d<^ Jo - ^^ ^ I {^ + cos (f> {f - l)*}~-« 8in»^^ <^1 + ^^ I {-g: + cos <^ (-?» - l)*}»-*»(2m -1) 8in*^»<^ cos ^d<^, 234 TRANSCENDENTAL FUNCTIONS. [CHAP. X* "^ 2m ?:±^ ['[z + cos A (^ - 1)*}'-^ sm«~ <^d<^ Zm — 1 J = rsin««-« <^ {ir + cos (z* - !)*}*-« {1 + r (-r« - 1)-* cos <^} d<f> Jo = — — - -J- \{z + COS 6 (z^ - l)i}«-"»+i 8in«*-» <bd6. n — m + 1 d^^Jo T-v /) Y' r Thus if we write Tm-l (2: + cos (^ - !)*}'*-« sin** <^c^, we have /„=_-(2^1) %> , (n + m)(n— OT+1) dz and therefore I - (2m- l)(2m-3) ... 1 ^ ** (n + m)(n + m-l)...(n-m + l) cJ^** But /o = f '{^ + cos <^ (^» - 1 )*}*» d<^ = irPn {z\ Jo when the real part of z is positive. Therefore /. = , (2m -1) (2m -3) 1.^ ,, ;^Pn(.) = (2m-l)(2m-3)...1.7r ^ -^ ^ (n + m)(nH-m-l)... (n-m + l)^ '^ "^ ^ ^' """^ ^~ ^^^^ (2m-l)(2m-3)...1.7r ^^"^^ X I {-^ + COS <^ (^« - 1)*}»-^ sin*** ^ d^. Jo This result expresses Pn^(z) as a definite integral of Laplace's type, valid for all values of n when the real part of ir is positive. 132. Alternative expression of PfJ^ (z) as a definite integral of Laplace* 8 type. The formula last found can be replaced by another result, found in the following way. If in Jacobi's well-known theorem* Jy (co8^)co8m^d^ = 1.3.5 ..\2m-l) /o'-^'"'^^^"*>°^°"'^'^- we take /(cos ^) = {^ + (z^ — 1)* cos <f>}^, * CreUe's Journal^ xv. H 132, 133] LEGENDRE FUNCTIONS. 235 80 that /w (cos <^) = n (n - 1) ... (n -m + 1) (z^ - 1)^ {2:+ (^ - 1)» cos <^}"^, we obtain I {^ + (^— l)*cos^}**cosm^d^ Jo ^ njn-l) ... (n-m-hl) , _ , .f 1.3.5...(2m-l) ^ ^ X I {z+ (z^- 1)* cos 0}»*-^ sin^ <f)d<f> Jo (n + m)(n +m— 1) ... (/i + 1) Therefore p^m /^\ ^ (^ + m)(n4-m-l) ... (n + l) , ^vf TT X 1 {2^ + (^ — 1)* cos (^J** cos m<f>d(f>. Jo This formula is valid for all values of w, and for all values of z whose real part is positive ; m being a positive integer. 133. The function Cn*" («). A function connected with the associated Legendre functions F^^ {z) is the function On*" (^)) which for integral values of p is defined to be the coefficient of A** in the expansion, in ascending powers of A, of the quantity (l-2A^+A«)-»'. It is easily seen that C^*" (z) satisfies the differential equation <Py (2v + l)zdy n(n + Qv) d^^ z^-\ dz z^-l ^""* For all values of n and y, it may be shewn that C^ {z) can be defined by a contour- integral of the form Constantx (1 -««)*- ^ f ^lzf)-l^dt. When n is integral, we have Cy(z^= (-2)*v(i^4-l)...(v4-n-l) * ^ ^ n! (27i+2v-l)(2»+2v-2)...(n+2i^) which corresponds to Rodrigues' formula for P^ (z) ; in fact, since F^(z) = Cj{z\ Bodrigues' formula is a particular case of this formula. 236 TRANSCENDENTAL FUNCTIONS. [CHAP. X. When r is an integer, we have «-'*^^^"(2r-l)(2r-3)...3.1 (£2^ *^^^' whence we have The last equation gives the connexion between the functions C^^ (z) and P/ (2). This function C/ (z) has the following further properties, analogous to the recurrence- formulae, <t;(.)-c.(^)-£c;w=o. Ow-<l«="-:^c'/(.). r-i Miscellaneous Examples. 1« Shew that when n is a positive integer, p^{z)J-z:^-p(u^+z^r^, where t^* is to be replaced by (1 -^) after the differentiation has been performed. 2. Prove that when w is a positive integer, (Cambridge Mathematical Tripos, Part I, 1898.) 3. Shew that • -1 w « fl.3.6...(2w-l)l«,„ .. „ ... (Catalan.) 4. Prove that is zero unless m- n«i ± 1, and determine its value in these cases. (Cambridge Mathematical Tripos, Part 1, 1896.) 6. Shew (by induction or otherwise) that when n is a positive integer, (2»+l)rP,*(«)(fc-l-*P,«-2i(P,«+i','+"+^»-t)+2(A'P»+^»^»+-+^»-i^«) (Cambridge Matbematdcal Tripos, Part 1, 1899.) /- 6. Shew that, if i{r is an odd number, 1 — jfe-2a»P»(4 (1-22A+A«)« MISC. Exa] LKOENDRE FUNCTIONa 237 where 8 a \i(t-») _A« 2^*-»(2n+l) / 8 3\ " (l-A«)*-«1.3.6...(it-2)V 8a? 8j^; ^ ^-i(2i»-t+4)yi (2n+Jfc-2)^ where a: and y are to be replaced by unity after the differentiations' have been performed. (Routh.) 7. If n-O shew that and and where 2 (n+1) 72»+i -3 (2n+l) /?^+(2n- 1) i2»_2=0 4(4a3-l)/J^'"+96««72»"-«(12n«+24w-91)i2^'-w(2n+3)(2n+9)i2»-0, K-J^ , etc (Pincherle.) 8. If m and n be positive integers, and m^n, shew that P (z)P (z)== 2 A„,^rArA^.r ( 2n+2m-4r+l\ where ^«' 1.3.5... (2m- 1) ml (Adams.) 9. Shew that P^ (z) can be expressed as a determinant in which all elements parallel to the auxiliary diagonal are equal (i.e. all elements are equal for which the sum of the row-index and column-index is the same) ; the determinant containing (2n- 1) rows, and its first row being (Heun.) _1 1 11 ^ 3' 3''-"5' 5^'"*2w-l t2. 10. Shew that 11. Shew that 2 r {z(l-i^-2t(l^m^ ^^^ 12. Shew that, when n is a positive integer, §^(cos^) (-1)** 8» fl f^tn-*-! n! 8«* (2V*«(r-+i)}' where ««=r cos ^. (SUva.) (Catalan.) 238 TRANSCENDENTAL FUNCTIONS. [CHAP. X 13. Shew that the complete solution of the Legendre differential equation is y=^p.(.)-H5P.w/;^3-^^^,. 14. Shew that {2 + (2i^l)i]a^ 2 5«§2m.«.lW, where „ o(a+2 w+i) r(m~^)r(m-a-^) "*" 2tr ml r(m-a+l) * •'x+(««-l] dh. 15. Shew that, when the real part of (w+ 1) is positive, and /-^-(^t-i)* A** 16. Prove that (Cambridge Mathematical Tripos, Part II, 1894.) 17. Shew that, if n be a positive integer, 18. Shew that and where n is a positive integer, and z > 1, and where log ^ is to be changed into log z— 1 \ —z if « is numerically less than unity. Prove also that V 2 3/ P223« V 2 / ■*"••* whereit=l + - + 5+...+-. 2 3 n (Cambridge Mathematical Tripos, Part II, 1898.) MISC. EXa] LEGENDRE FUNCTIONS. 239 19. Shew that -P«*»(«)=-^ ^-^, ^f^F{n, n+m, m+1, s^). 20. Prove that, if ^•"n (n«-l)(n2- 4) ...{»«-(«- 1)2} (w+«)^^ ^^ ^» then ,. -P 2^2n + l) 2n+3 3 (271+3) p , 3(2n+5) ( 2n+3)(2n+ 5) y3"^-+8 2«-l ^»+i"^ 271-3 *-i (2w-l)(2w-3)^'-5' and find the general formula. (Cambridge Mathematical Tripos, Part II, 1896.) 21. If shew that 22. If On"" {^^1 - (^ - 1)* (^1^ - 1)* cos <^} _ n(2y-2) J^-* 4An(n-X){n(v+X-l)}g {n(y-l)}2xfo^ ^^ n(n+2i.+X-l) (Qegenbauer.) <^» («) = I **(^- 3^2 + l)-i i»c?<, where ^i is the least root of ^— 3te+ 1 =0, shew that (27i+l)crn+i-3(2w-l)z(r„.,+2(n-l)(rn-2=0, and 4(4«5-l)(r,,'"+1442j<r,»"-«(12w2-24n-291) <r,»'- (n-3) (2n-7) (2n+5) cr,»=0, where o-n' = - ^} , etc. (Pincherle.) /'/ 23. Shew that (Hobson.) g^n>(,)^^>i r(.^+l) r COShj^t. r(n-w + l);o {r+(22-i)icoshw}-+i where the real part of (n+l) ia greater than m. 24. The equation of a nearly spherical surface of revolution is r=l+a{P,(cos^)+P3(co8^ + ... + P2n-i(cos^)}, where a is small ; shew that to the first order of a the radius of curvature of the meridian is l+a 2 {n(4wi+3)-(r?i+l)(8m + 3)}P8,n+i(cos^). (Cambridge Mathematical Tripos, Part 1, 1894) CHAPTER XL Hypbrgeometric Functions. 134. The Hypergeometric Series. We have already in § 14 cjonsidered the hypergeometric series* a.b a(a + l)b{b + l) a(a + l)(a + 2)b(b + l)(b + 2) ■^l.c "^ 1.2.c(c+l) "^ 1.2.3.c(c-rl)(c + 2) "^ "' from the point of view of its convergence. It was there shewn that the series is absolutely convergent for all values of z represented by points in the interior of the circle whose centre is at the origin and whose radius is unity. It follows from §22 that all the series which can be derived from the hypergeometric series by diflferentiation and integration are likewise absolutely convergent within the same region : and by § 55, the convergence is not only absolute but uniform over the interior of the circle, and the sums of the series obtained by differentiation and integration of the series term by term are the derivates and integrals respectively of the sum of the series. The hypergeometric series, together with the series which can be derived from it by the process of continuation (§ 41), will therefore represent an analytic function of the variable z; this function will be denoted by F(a,b,CtZ), Many of the most important functions of Analysis can be expressed by means of the hypergeometric series. Thus it is easily seen that (l+zY^F(^n,/3,/3,^zl log(l+5) = zi^(l,l,2,-5), 6^ = Limiti^U,/8,l,j), and we have shewn in the preceding chapter that the Legendre functions may be represented by the series • The same was given by Wallis in 1655. 134, 135] HTPBRGEOMETRIC FUNCTIONS. 241 2-z-r(n-^i) (l^n n 1 1\ ^«W- ^4r(n + l) ^\ 2 ' ""2' 2"^"' J«;' O /.,x_ 7r>r(n + l) 1 y/n+1 7i + 2 3 1\ These examples are sufficient to shew that the functions represented by the hypergeometric series are in some cases one-valued and in other cases many-valued. Example, Shew that ^^F(a, 6, c, «)=^F(a + l, 6+1, c+1, z). 136. FaZt^ o/* ^A« series F(a, 6, c, 1). We have shewn in § 14 that the series F(a, b,c,l) converges absolutely so long as the real part of c — a — 6 is positive. Suppose this condition to be satisfied. Then we have F(nhr^\-l. Tia + n)T(b + n)r(c) F (a, 6. c, 1) ^ Z^ ^uT(c + n)r(a)r(6) Tic) I 1 T{n + a)T{n-\-h)Vic-h) r(a)r(6)r(c-6),tonI r(n + c) r(c) I ir(» + o)5(n + 6,c-6)L r(o)r(6)r(c-6),.on! r(c) = r(a)Tl)l(c-b) iy'^''^IS^' - ^y-'-^^'dz. Writing z=^l — t, this becomes (writing xt^s) r(c) r(a)r(6)r(c-6);o Jo r(c) zrr\j '^1 e-'8^^if>-<'-<>-'(l-tf-\dt r(a)r(6)r(c-6) r(o)5(c-a-6,6) r(c)r(c-o-6) ~r(c-6)r(c-o)* W. A. 16 242 TRANSCENDENTAL FUNCTIONS. [CHAP. XL The hjrpergeometric series with argument unity can thus be expressed in terms of Oamma-functions. 136. The differential equation satisfied by the hypergeametric series. The function represented by the hypergeometric series y='F(a,b,c,z) satisfies the differential equation 'i^-l)2^ + l-o + ia + b + l)z]^ + ab!, = 0; for if the series be substituted for y in the left-hand side of the equation, the coefficient of 2^ is a(a + l)...{a + r-l)b(b + l)...(b + r-l) 1 .2 ...r.c(c + l)...(c + r) {r(r'-l)(c-{-r)'-r(a-\-r)(b+r) -c{a+r)(b + r) + r(c + r)(a + 6 + 1) + oft (c + r)} or zero ; which establishes the result. Example, Shew that one integral of the equation IS where ^F{m-/i, m-v, m-n+1, «), a-l--Oi+ir), 6+l=m+n, 137. The differential equation of the general hypergeometric Jimction. The differential equation found in the preceding article is a case of a more general differential equation, which may be written d2l' \ z — a z — b z — c ] dz { aa'{a^b){a-c) ^ /3^(b^a)(b^c) ^ yy-(c-a)(c-6) j y ^^ ( 2r — a z — b z — c J (^— a)(2r--6)(2r— c) * ...(A), in which a, 6, c, a, ^, 7, a , ff, y are any constants such that the equation a + /3 + 7 + a' + )S' + 7' = l is satisfied. This will be called the differential equation of the general hyper- geometric function. The form here given is due to Papperitz*. * Math, Annaleuj zxv. 136, 137] HYPEEGEOMETRIC FUNCTIONS. 248 We shall now shew that the differential equation satisfied by the hyper- geometric series is a particular case of this equation. For in the equation (A), write = 0, 6=00, c = l. The equation becomes y . f l-g-g' . I-7-7 ) dy ( ««' Tx' . oo/l _y_ n In this equation, let a and 7 be replaced by zero. We thus have d2^ \ z z—l J dz ziz—l) and in this equation the constants ol, y, fi, ^, are to be such as to satisfy the relation /g + a' + ZS' + y^l. This differential equation can be identified with the equation z(z-l)^ + {-c + (a + b + l)z]^ + aby = 0, which is the differential equation satisfied by the hypergeometric series, by writing y3 = a, )8'=:6, a' = l-c; which in virtue of the above relation gives 7' = c — a — 6. The differential equation of the hypergeometric series is therefore a special case of equation (A). We shall denote any solution of the general differential equation (A) by the symbol {a b c a ^ y z}. a! ^ 7' This notation is due to Riemann* ; it enables us to express our result thus : The hypergeometric aeries F(a, 6, c, z) is a solution of the differential equation of the class offu/nctions t 00 1 P a z [ 1— c b c—a—b * Abhandlungen d, K, Getell, d, WUtemchaften zu OdtHngen^ Tn. (1857). 16—2 244 TRANSCENDENTAL FUNCTIONS. [CHAP. XI. Although the hypergeometric series itself satisfies only a particular form of the differential equation (A), it is nevertheless possible to satisfy the general equation (A) by means of a function derived from the hjrper- geometric function. For by the transformation a? (5 — 6) (c — a) = (^r — a) (c — b), the differential equation (A) is reduced to the form cte* I X . a?— IJow? ( X a? — 1 } x{x — l) In this we take a new dependent variable, defined by the equation y = x*{l'-x)y u. The equation becomes , /I -a' + a , 1 -7' + 7\ du . ,^^ . x/i ^ /\> ^ /v Now the equation a + /S + 7 + a' + ^' + 7'sxl will be satisfied if j3, a\ ffy 7', are expressible in terms of three new constants^ a, 6, c, defined by the formulae / /8 = a — a — 7, a' = 1 — c + a, ^ = 6-a-7, 7'=sc — a — 6 + 7. The differential equation for u can now be written d^u du x(x- l);i-i + {(1 +a + 6)a?-c}-Y- + a6M=0. But this is the differential equation satisfied by the hypergeometric series, a solution of it being F (a, 6, c, x). Hence we have, as one solution of the equation, u^F(a + /3 + y, a + /9' + 7, 1 + a - < a:), or y^x*(l- x)yF(a + ;8 + 7, a + /S' + 7, 1 + a - a', x), or, disregarding a constant factor, 1 • 138] HTPERGEOMETRIC FUNCTIONS. 245 This is therefore a solution, expressed by a hypergeometric series, of the differential equation which defines the class of functions Ia b c a ^ y z The advantage of the differential equation (A) over the equation found in § 1 36, which is satisfied by the hypergeometric series, lies in its greater symmetry and generality. The points 2r = a, -e = 6, and z=^c, are called the singularities of the differential equation (A); the quantities a and a' are called the exponents at the singularity a; and similarly /8 and ^ are the exponents at h, and 7 and y' are the exponents at c. Example, Shew that 00 1 ^ / -1 00 1 fi y zn^p\ y 2/3 y k »' y' ] I y s/y y (Riemann.) This relation follows from the fact that the differential equation corresponding to either of the P-fiinctions is 138. The Legendre functions as a particular case of the hypergeometric function. The expressions which have been found for Pn(^) and Qn(^) as hjrper- geometric series naturally lead us to suppose that Legendre's differential equation is a special case of the differential equation which defines the general hypergeometric function. That this is the case appears from the following investigation. If in equation (A) of the last article we take a = — 1, 6 = 00, c = l, we obtain the differential equation d»y ( l.,a^g^ 1-ry^ryO dy r 2ga^ 277M y d?^[ 5 + 1 ^ 5-1 ]dz^\ 5+l'^^^"^5-l](5-l)(5 + l) ''• If now in this equation we take a = 0, a' = 0, 7 = 0, ♦/ = 0, yS = n + 1, fi'ss^n, we obtain or (l-^')§-25^ + n(n + l)y = 0, which is the Legendre differential equation. 246 TRANSCENDENTAL FUNCTIONS. [chap. XI. It follows from this that any solution of Legendre's equation is a hyper- geometric function of the type -1 00 1 p\ n + 1 z -n In the same way it can be shewn that the associated Legendre functions Pn^{z) and Qn^{z) are hypergeometric functions of the type P\ -1 00 2 " + 1 1 m \ I m m "2 -" "2 Example 1. Shew that S^-«=^ -1 00 — r w+r+1 -n^r 1 \ -r z • • Example 2. If «^si7, shew that the Legendre difierential equation takes the form c&,«"^\2,,"l-i7Jrfi7"^ 47(1-1;) Shew that this is a hypergeometric differential equation. 139. TransformcUioris of the general hypergeometric function. We shall next consider the effect of performing certain transformations in connexion with the general hypergeometric function ^a b c The differential equation satisfied by this function is d^ \ z^a z — b z — c)dz\ ~ ' z — a P^{b^a){b^c) . 77'(c-a)(c-6) z^b z — c \ y_ ){z- a)(z - b)(z - c) = 0. In this equation, let the dependent variable be changed by the trans- formation 139] HTPERQSOMETRIC FUNOTIONS. 247 \ The differential equation for y' is found after a slight reduction to be <fy ^ f l-«-«'-28 ^ l-/3-/y + 28 1-y-y' ) dy" dz* \ z — a z — b z — c)dz ^ f (a + 8)(a' + 8)(a 1 z ^a - 6)(a - c) ^ (/3 - 8)(/y - 3)(6 - c)(6 - a) + 77 ,(c~a)(c-6) z — c I (-P — a) (^ — b)(z — c) ^-6 = 0. This is the differential equation of a hypergeometric function which has exponents a + S, a' + S, at the singularity a, and exponents )8 — S, )8' — S', at the singularity 6 ; and so we have C-5 -./T.\« a" = P- a 6 c a + S )8— S 7 z a' + S /y-s y and hence in general we shall have (-;)'&:)■' I a b c . a ff 7 = P a a + S id- a' + S /3' b c -S— € 7+6 — S — € 7' + € It will be observed that by this transformation the exponent-differences a — a', )8 — )8', 7 — 7' are unaltered. Consider now the effect of transformations of the indepeAdent variable s. If we introduce in place of jer a new variable z\ defined by the equation 5= Oi^ + ^i C^ + di' where Oifb^ Ci, di are constants, so that we have , - di^r + 6, z = and dy _ Oidi - 61C1 dy ^ (c^z + d^ydy dz (CiZ — Oi)* dz' Oidi — 6iCi de' rf^*^ ((hz-ihT dz {CiZ-OiY d/* ^ 2c(c,z' + d,y dy {ci^ + d^Y d^ 248 TRANSCENDENTAL FUNCTIONS. [chap. XL Hence if we define quantities a', b\ c by the relations __ _ _ OiC + 61 Cia' + di * Cjjb' + di * so that the general hypergeometric differential equation becomes [9. . (l-a-aO(Cxa^ + d,) , (1 -/3 - ^yXc^ + d,) z — a + a-7"-70( 77'(o'"-a')(c'-6')) y 7T7T7-:7x=0. The coeflBcient of -7^ in this equation can be written in the form l-.a-a^ l-/8~)y 1-7- 7 , 1 which, in virtue of the relation 2ci-(l-a-a')Ci \ reduces to a + a' + /3 + )8' + 7 + y = l, i^g^g- i^ff^/y i-7^y Hence the differential equation reduces to the differential equation of the function a c p\a fi y z a' /3' 7' and thus we have the relation a 6 c P-la fi y z a' ^ 7' o =P- 6' c' ^ 7 ^' ^ 1' This shews that the general hypergeometric function is unaltered if the quantities a, b, c, z are replaced by qualities a', b\ c\ z^, which are derived from them by the sam^e homographic transformation. 140] HYPERGEOMETRIC FUNCTIONS. 249 140. Ths twenty-four particular solutions of the hypergeometric differential equation. We have seen in § 137 that a particular solution of the general hyper- geometric differential equation is <z ^r C-^)V f ./..,... /r ^ ,-, 1 .. - .'. I^^fe^j)} . We shall suppose that no one of the exponent-differences a — afy — ^, 7 — 7' is zero : it is shewn in treatises on Linear Differential Equations that when this exceptional case occurs, the general solution of the differential equation involves logarithmic terms; the formulae will be found in a memoir* by Lindelof, to which the reader is referred. Now if a be interchanged with a', or 7 with 7', in this expression, it must still satisfy the differential equation, since the latter would be unaffected by this change. We thus obtain altogether four expressions for which (c — 6) (^ — a) (c — a) (-8^ — b) is the argument of the hjrpergeometric series, namely .fz-ay' /z-c\y „{ , r, , OP / -, . / (c - 6) (-gr - a)' these are all solutions of the differential equation. Moreover, the differential equation is unaltered if the quantities a, a', a are interchanged respectively with ^, ff, 6, or with 7, 7', c. If therefore we make such changes in the above solutions, they will still be solutions of the differential equation. Let a change in which (a, a', a) are interchanged with ()8, /S', 6) be denoted for example by /a, 6, c\ U, a, 0) ' each singularity in the bracket being interchanged with the singularity above or below it. Then there are five such changes possible, namely, (a 6 c\ fa h c\ (a h c\ (ah o\ (a b c\ \b c a)* \c a bj* [a c b) ' [c b a)' \b a cj' * Acta Soe, Scient. Fennieas, xix. (1898). 250 TRANSCENDENTAL FUNCTIONS. [chap. XL To each such change correspond (by interchanging a with a, etc. as already explained) four new solutions of the differential equation. We thus obtain twenty new solutions, which with the original four make altogether twenty- four particular solutions of the h3rpergeometric differential equation, in the fqrm of hypergeometric series. The twenty new solutions may be written down as follows : -c)j +y8-)8', + /3'-/9, + /3-/9', + /8'-/8. + 7-7', (a - h) (z {a-c){z-h)\ (a-b)(z-oX (a -c)(z — b) -6)1 -c)\ (a - b) (z (O — C) (^r — 6) (6- (6 - o) (g - c) | - c) (^ - a)j + V-~ (^-»)(^-g) t + 7 7. (6_c)(^_a)[ (6 — o) (« — c) + 7-7'. + 7'-7» (6 - c) (^ - o), (6 — a)(^ — c) (fc_c)(^-aX „_„' (fe-c)(^-a )) "' (6-a)(^-c)J (6-c)(ir-a) ] ' (6-a)(^-c)J Q>-c){z-a) \ (6-a)(^-c)[ (6-c)(^-o) ] (6 — a) (5 — c) (a-c)(-? (a-6) (£- Xa — c^iz + 'y "y' {fl-c){z-b)\ (a-6)(g-c)) -6)f + a-a', + a'-a, + 7-7'. + 7' -7. -6)1 + y'-% (a — c)(z 141] HYPERGEOMETRIC FUNCTIONS. 251 The existence of these twenty-four values was first shewn by Kummer* Example. Find the twenty-four solutions of the L^endre differential equation, corresponding to the above set of solutions of the hypergeometric differential equation ; and express each of them in terms of the two independent solutions P^ {z) and Q^ (z), 141. Relations between the particular solutione of the hypergeometric diferential equation. Since the twenty-four expressions found in the last article are solutions of the same linear differential equation of the second order, any three of them must be connected by a linear relation with constant coefficients. We proceed to find the relations which thus connect them. First, consider the set of four solutions Vi* Vii Viz* yi6> it is clear that, in the neighbourhood of the point z=^a, each of them can be expanded in a power-series of the form 4 (2r - a)* {1 -h £ (^r - a) + (7(z - a)» + ...}. But there is only one series of the form (z-a)* {1 -h£ (^r- a)-f C(2r- a)» + ...} which satisfies the differential equation; for the coefficients 5, C, ... can be uniquely determined by actual substitution in the differential equation. Let this solution be denoted by P<*>. Thus the solutions t/u Vty yi8, yi5 must be mere multiples of P^*'. Moreover, for y, the factor A \s (a - c)y {a - 5)-c+y> ; for y, it is (a - c)y\a - 5)-<«+y'» ; for y^ it is (a — 5)^ (a — c)~*""^ ; and for y„ it is (a — bY{a — c)~*~^'. * Crelle*s Journal^ xv. 252 TRANSCENDENTAL FUNCTIONS. [CHAP. XI. Thus we have ^ ^ Ka — cl xa — hl xi-ja + ZS-K,. « + ^' + y. l-,„-a'.J--|(i^)} \a — c' \a — - 6\-"-r' 6, x^{«+^+y, „+^+,. !+«-«, ;::^;;;:g } =(-«)• (src^D xP|a + ^ + -y. a + /9+r. l + a-«. (^ _ a) (^ - c) , ■?■ fz-h\l>' =<->-c^r(^^) x'h^*''. ■>+«+»', 1+.-.', fti:^';::} }- Similarly solutions P<*\ P<^», P<^\ P^y\ P<y'> exist, each of which is equi- valent to four of the above hypergeometric series. Having thus classified the twenty-four solutions into six distinct solutions, namely p(a)^ p(e')^ pO)^ p(/r)^ p(y)^ p(y')^ we proceed to find the relations between these latter six solutions. We know that P'*> must be expressible linearly in terms of P<y> and P^y'K Let the relation between them be P(«) = a^p(y> + Oy^pty). We have then to find the coe6Bcients Oy and oy. Now this equation can be written in the form r{'*»^i. '*g*i: '+«-«'. I^^Jl z - 5\-»-r -('-)' (j^ro f |.^^^„ ^,^^„ ,^,.,., («_^|g^i 5-6N— *' 142] HYPERGEOMETRIC FUNCTIONS. 253 Dividing throughout by the common factor (z — a)*, and writing ^ = a and z=sc successively in the resulting equation, we obtain two equations, from which 7y and oy can be found : the hypergeometric functions reduce to the type F(u, V, w, 1), which in § 135 was shewn to be expressible in terms of Gamma-functions, and the type F {Uy v, w, 0), which clearly has the value unity. As already explained, in certain cases (e.g. when one of the exponent-differences is an integer) the above theory of the solutions requires modification. For a discussion of these cases the student is referred to Lindel6f s paper already mentioned, and Klein's Lectures " Ueber die hypergeometrische Function." 142. Solution of the general hypergeometric differential equation by a definite integral. We next proceed to establish a result of great importance, relating to the expression of the hypergeometric function by means of definite integrals. Let the dependent variable y in the differential equation of the general hypergeometric function ((A) of § 137) be replaced by a new dependent variable /, defined by the relation y^iz-aYiz-hf (z-c)y I. The differential equation satisfied by / is easily found to be 1 4-/9-/8' . 1 + 7-7] d*/ fl+a-a l4-i8-/8' dz^ ( z^a z — o z — o dJ dz (a + /3 + 7){(tt4-/3 + 7 + l)^ + Sa(a + /y-fy-l)} "^ {z-a){Z''b){Z'-c) which can be written in the form i (LJMi_l) p. (^) + (f _ 1) ^> (^) /. where { f = 1 -a-/3-7 = o' + /3' + 7', Q{z) = {z-a)iz-h){z-c), R(z) = 2 (a + y3 + 7) (z -b)(z- c). It must be observed that the function / is not regular at e.'oo , and consequent! j the above differential equation in / is not a case of the generalised hypergeometric equation. We shall now shew that this differential equation can be satisfied by an integral of the form 254 TRANSCENDENTAL FUNCTIONS. [CHAP. XI. provided the path C of integration is suitably chosen. For on substituting this value of / in the differential equation, the condition that the equation should be satisfied becomes = f (^ - ay+P+y' (t - 6)*+^'+r-i (t - cy-^P+y'-i {z - ^)-«-^-y-2 xdt, J c where + {t-z)[R{z) + {t-z)R{z)] = {i-^)[Q(t)-{t-zy\ + it-z){R(t)-{t-zr^(<^ + p + i)] ^{^-^)Q{t) + {t-z)R{t) = -(l + a + /3 + 7)(<-a)(<-6)(<-c) + 2(o' + /3 + 7)(<-6)(«-c)(«-«), or Z = (< - a)'-"'-*-y (« - hy-^-^-y (t - cf-^-f-y' (z - <)«-H>+y+« ^ {(« - aY'+^+y (t - by+^'+y (t - c)'+p*y' (t - ^)-o+«+p-h')). It follows that the condition to be satisfied reduces to dV o-lis"-!/^- where F= {t - a)*'+^+y {t - 6)«+^'+> (< - c)*+^+>' (^ - 5)-ti+«+^+y). The integral / will therefore be a solution of the differential equation, provided the path of integration C is such that the quantity V resumes its initial value after describing the arc C. Now F= (t - ay-^^-^r-^ (t - 5)*+^'+>-i {t - c)»+^+>'-^ (z - 0"^"^"^ ^, where U^it-a) (t- b) (t - c) (^ - 1)'^ ; and the quantity U resumes its original value after describing any contour : hence if (7 be a closed contour, it must be such that the integrand in the integral / resumes its original value after describing the contour. Hence finally any integral of the type {z^aY{z^hy{z''C)y f (t-ay-^-^^'-'(t-b)y-^'^P''Ht-cY-^-^'-'(z-t)'^-fi-ydt, J c 142] HTPERGEOMETRIC FUNCTIONS. 255 where C is either a closed contour in the t-plane such that the integrand resumes its initial value after describing it, or else is an arc such that the quantity V has the same value at its termini, is a solution of the differential equation of the general hypergeonietric function. Example 1. As an example, we shall now deduce a real definite integral which (for a certain range of values of the quantities involved) represents the hypergeometric series. The hypergeometric series F(a, by c, z) is, as already shewn, a solution of the differential equation of the function ' 00 1 P ■ a t 1-c b c^a-b The int^ral thus becomes in this case I <«-«(<-l)«-*-i («-«)-« eft. Now the quantity F is in this case and this tends to zero at ^b 1 and «= oo , provided c> b>0. Hence if these conditions are fulfilled, we can take as the contour C an arc in the ^-plane joining the points t^l and <nao ; so that a solution of the differential equation is / 00 f^ if- l)«-*-i {t-z)-^dt. 1 In this integral, write t^-; the integral becomes / this integral is therefore a eolutum of the differential eqtiatumfor the hypergeometric series. It is easily seen that this integral is in fact a mere multiple of the hypergeometric series jP(a, 6, c, z) ; for supposing | « | < 1, and expanding the quantity (1 - uz)'^ in ascending powers of z by the Binomial Theorem, the integral takes the form fV»(l-«)'-^»du+ i «(«+l)-(«+*-l) ^[V-'*r(i_^)«-t-.rf,^ Jo r=l T' Jo or r«i ri 256 TRANSCENDENTAL FUNCTIONS. [chap. XL or or nth . m/io. ; a{a'\-l)''>{a+r-\)b{h+\).,.{h'\-T-\) \ a(a+l)...(a+r-l)6(6+l)...(fe-hr~l) r!c(c+l)...(c+r-l) 5(6, c-6)jP(a,6, c,z), which establishes the result stated. Example 2. Deduce SchUiflUs integral for the Legendre functions^ as a case of the general hypergeometric integral. Since the L^;eQdre equation corresponds to the hypergeometric function -1 00 1 ^ n+1 £ V, . -n J the corresponding integral is or [ (^-l)«(«-0)-'»-icfe, taken round a contour C such that the integrand resumes its initial value after describing it ; and this is Schlafli's int^ral. Example 3. Deduce Laplace's integral for the Legendre functions, as a case of the general hypergeometric integral. If we write «=i(«*+r*), 4 the Legendre differential equation becomes ^-i-P I ^ \^y n(ti+l) y This corresponds to the hTpergeometric function .0 00 1 n "2 n + 1 2 f - y n+1 2 n 2 1 and so the hypergeometric integral becomes in this case ("i I tt» ( 1 - u) "* (i - w)"* du, taken round a contour enclosing the points u=l and u^(. Write Then the integral becomes taken round a contoiu* enclosing the points u^l and u^CK 143] HYPERGEOMETRIC FUNCTIONS. 257 Write u=sA( in this integral ; we thus obtain (l-.2«A+A«)-U»cM, /' the integral being now taken round a contour in the A-plane enclosing the points k=( and Suppose now that the real part of z is positive ; and let the contour become so attenuated as to reduce to a small circle surrounding the point h=(^ another small circle surrounding the point h=(~\ and the line joining the points ( and f "^ described twice. The small circles contribute only infinitesimally to the integral, which thus becomes a multiple of / ^^(1-2M+A«)-U«(i^ Writing A =«+(«•- 1)* cos in this integral, we obtain /; {«+'(««-l)*C08<^}*rf<^, which is one of Laplace's integrals (§ 119). 143. Determination of the integral which represents P<*^ We shall now shew how the integral which represents the particular solution P<*^ (§ 141) of the hypergeometric diflferential equation can be found. We have seen (§ 142) that the integral 7=(^-a)«(^-5)^(^-c)vf (^-a)^+y+«'-H«-6)y+*+^'-H«-c)*'^^'-K'^-0"*"'^"^* J c satisfies the diflferential equation of the hypergeometric function, provided (7 is a closed contour such that the integrand resumes its initial value after describing C, Now the singularities of this integrand in the ^plane are the points a, b, c, z; and on describing a simple closed contour enclosing the singularity 6 alone, the integrand resumes its initial value multiplied by I Jiniy+a+fi'-l) as is seen by writing it in the form (^+y+tt'-l)log«-a)+(y+a+/3'-l)log(f-6)+(«+/8+y-l)log(<-c)-(a4-/3+Y)log(«-0 e . Take then a point in the ^-plane, and draw a loop in the ^-plane passing through and encircling the point 5, but not encircling any of the points a, c, z. Let an integral taken in the positive or counter-clockwise direction of circulation round the perimeter of this loop be denoted by the sign Jo • W. A, 17 258 TRANSCENDENTAL FUNCTIONS. [CHAP. XI. and let an integral taken in the negative direction of circulation round the perimeter of the loop be denoted by r(6-) so that we have the equation /•(&+) r(b-) Jo Jo where it is understood that the initial value of the integrand in the second integral is taken equal to the final value of the integrand in the first integral Let now a contour C be drawn in the following way. Take first a loop starting from 0, encircling the point 5 in the positive direction, and returning to ; then a loop starting from 0, encircling the point c in the positive direction, and returning to 0; then a loop encircling the point 5 in the negative direction ; and lastly a loop encircling the point c in the negative or clockwise direction. Conformably to the notation already explained, an integral taken round this contour will be denoted by {b+, c+, 6-, C-) Jq Now after description of this contour, the integrand of the integral / already considered resumes its initial value multiplied by gaIrf(y+tt+/8'-l-hE+^47'-l-y-«-^'+l-«-^-y'+l) or 1, i.e. the integrand resumes its initial value*. Hence if C be taken as the contour, the integral / will satisfy the differential equation. Thus /•(6+,c+.6-.c-) J = (^ - aY (z -hfiz" c)y (t - ay+y +«'-' (t - 5)y+-+^'-^ Jo satisfies the differential equation of the hypergeometric function. Now suppose that the point z is taken near to the point a, so that |z — a| is less than either |6 — a| or |c — a|. We can clearly draw the contour just * These double-circuit inUgraU were introdaced by Jordan in 1887. Clearly any namber of contours can be formed in this way, it being necessary only to ensare that each singular point is encircled as often in the negative or clockwise direction of circulation as in the positive or counter- clockwise direction. 144] HTPEBQEOMBTRIC FUNCTIONS. 259 described in such a way that, for all points ^ on it, I ^ — a | is greater than | ^ — a | . Thus we can write Jo (. . ^-, (, - izf) z - a\-*-^-y dt Under the conditions already stated, each of the expressions can be expanded by the Binomial Theorem in ascending powers of (z — a). We thus obtain for I an expansion of the form / = (2r - a)« {4 + £ (ar - a) + (7 (^ - a)« + . . . }, and as / satisfies the differential equation it must therefore be a multiple of the particular solution P**^ of § 141. Thus pw = Constant x (^ - a)« (z -hfi^z- c)^ {t - af-^y^'"^ Jo (t - 6)y+«+^ -> (t - c)-^+y'-i (z - 1)-^-^-^ dt Similarly r(6+.c+,6-,c-) P^*') = Constant x (z - a)*' (z ^hf{z- c)y {t - ay+y+— ^ In the same way the particular solutions P^\ P<^'»,*Pw, P<y'>, can be expressed as contour-integrals. 144. Evcduation of a dovhU-contour integral. We may note that an integral /; can be expressed in terms of the integrals in the following way. Let the initial value of the integrand at the point be denoted by T. After describing the loop roujid a, the integrand will have at the value g^'^^^+^+V"^) Ty and the part 17—2 ■(o-»-,6+, o-,6-) 260 TRANSCENDENTAL FUNCTIONS. ' [CHAP. XI. I of the integral I will have been obtained. Describing next the loop /(o+, 6 + . 0-, 5-) will therefore be ^»i(a+/r+y- ■'/r*'' and the integrand will return to with the value ^ (tt'+/8+y-l+e+^'+y-l) qt Describing next the loop round a in the negative direction, we observe that the corre- sponding part of the integral would have been r: if the integrand had had for initial value which is its final value when the loop is described with the initial value ^ : it is therefore actually or — e jo ' i. ' _ 21^i(«+^'+y-l) and lastly, describing the loop round h in the negative direction, we obtain the part of the integral. Collecting these r^ults, we have /■''•+'*+'»-'*-L(i-«««t«+c'+y-i>) /■'"*'_(! _««'rt(«+o'+y-i)) /■•***, a formula which furnishes the value of the double-contour integral in terms of two simple- contour integrals. 146. Relations between contiguous hypergeovietric functions. Let P (z) be a hypergeometric function with the argument z, the singu- larities a, 6, c, and the exponents a, a',/8, I3',y,y. Let Pi+i^rn-i(z) denote the function which is obtained by replacing two of the exponents, I and m, in P(z) hy I + 1 and m —1 respectively. Such functions Pi+i^fn^i{z) are said to be contiguous to P (z). There are clearly 6 x 5 or 30 contiguous functions, since I and m may be any two of the six exponents. It was first shewn by Riemann* that the function P(z) and any two of its contiguous functions are connected by a linear relation, the coefficients in which are polynomials in z, » • Abhandlungen der Kdn, Qet. der Wiss. zu O&ttingen, 1867. 145] HTPERGEOMETRIC FUNCTIONS. 261 80 29 There will clearly be — ^ — or 435 of these i*elations. In order to obtain them, we shall take P (z) in the form P (z) = (^ - a)* (^ - by (z - c)y lit" df-^-^'-^ (t - 5)y+«+^'-i Jc (t - cy+^+y'-i (t - ^)— ^ dt, where C may be any closed contour in the ^-plane such that the integrand resumes its initial value after describing C, First, since the integral round C of the differential of any function which resumes its initial value after describing G is zero, we have = J ^ {(^ - ay-^f"^ (t - 6)-+/»'+r-i (t - c)«+^+y-» (t - r)-H»-y} dt, or = (a + ^ + 7) f (« - ay-^+r-' (t - ftV+^'+y-i (^ - cy-^P+y'-i (t - -?)-- ^-r dt Jo + (a + )8' + 7 - 1) f (« - a/'+^+T' (e - 6)*+/»'+y-« (« - c)*+^+r'-i (t - «)-«H»-y (ft + (a + )8 + 7' - 1) f (t - a)*'+^+y ft - 6)•+^'+r-l rt - cV+^+y-i (^ - ^)-Hi-y d^ - (a + )8 + 7) f (« - a)*'+^-h' (e - 6)*+^'+y-» (t - c)-^^+r -^ (« - ^)-Hi-y-i d«, or (a' + )8 + 7)P + (a + )8' + 7-l)P.'+i.r-., + (a + )8 + 7'-l)P.'+,.y-i (a-h^ + 7) p ^-6 /l+l.Y'-l. Considerations of symmetry shew that the right-hand side of this equation can be replaced by jer — c These, together with the analogous formulae obtained by cyclical inter- change of (a, a, a) with (b, /8, )8') and (c, 7, 7O, are six linear relations connecting the hypergeometric function P with the twelve contiguous functions P«+l.A'-l» P/M-1,y'-1> Py+1,«'-1> Ptt+l,y-l> Pi8+1,«'-1, Py+l.A'-l* Pa'+i.ir-i, P«'+i.y-i, P/i'+i^y-i, P/i'+i,.'-!, Py+i,.'-!, Py+i,/i'-i. 262 TRANSCENDENTAL FUNCTIONS. [CHAP. XI. Next, writing < — a = (^ — 6) + (6 — a), and using Pa'-i to denote the result of writing a' — 1 for a' in P, we have Similarly P = Pa'_i,y+i + (c - a) P.-i. Eliminating P.'-i from these equations, we have (c - 6) P + (a - c) P«'-i. fi^+, + (6 - a) Pa'-i.y+i = 0. This and the analogous formulae are three more linear relations con- necting P with the last six of the twelve contiguous functions written above. Next, writing (^ — jer) = (^ — a) — (jer — a) we readily find the relation P = ^j P/1+i.y.i - (^ - aY^' (z - b)f^ (z - c)y xf(t- ay-^-^'-^ (z - a)y+«+/»'-^ (z - 6)«+^+y-i (^ - ^)-«H»-y-i d^, which gives the equations {z - a)-» {P - (^ - 6)-^ J'/i-M.y-il = (^ - 6)-^ [P-i^' or' Pr+i..'-il - (^ - c)-^ {P - (^ - a)-> P.-M.^'-i}. These are two more linear equations between P and the above twelve contiguous functions. We have therefore now altogether found eleven linear relations between P and these twelve functions, the coefficients in these relations being rational functions of z. Hence each of these functions can be expressed linearly in terms of P and some selected one of them ; that is, between P and any two of the above fwnctions there exists a linear relation. The coefficients in this relation will be rational functions of z, and therefore will become polynomials in z when the relation is multiplied throughout by the least common multiple of their denominators. The theorem is therefore proved, so far as the above twelve contiguous functions are concerned. It can in the same way be extended so as to be established for the rest of the thirty contiguous functions. Corollary, If functions be derived from P by replacing the exponents a, a', )8, )8', 7, </, by a+p, «' + ?, )8 + r, /S^H-*, 7 + ^, 7 +ti,where p, 5, r, «, f, w, are integers satisfying the relation then between P and these functions there exists a linear relation, the co- efficients in which are polynomials in z. MISC. BXS.] HYPERGEOMETRIC FX7NCTI0NS. 263 This result can be obtcdned by connecting P with the two functions by a chain of intermediate contiguoi^s functions, writing down the linear relations which connect them with P and the two functions, and from these relations eliminating the intermediate contiguous functions. It will be noticed that many of the theorems found elsewhere in this book, e.g. the recurrence-formulae for the Legendre functions (§. 117), are really cases of the theorem of this article. MlSOELLANEOUS EXAMPLES. 1. Shew that c 2. Shew that jP(a+l, 6+1, c, t)-F(a, 6, c, t)^^F{a+l, 6+1, c+l, a). 3. If P(«) be a hypergeometrio function, express its derivates -^ and -^ linearly in terms of P and contiguous functions, and hence find the linear relation between P, -^ , cPP and -rj , i.e. verify that P satisfies the hypergeometric dififerential equation. 4. If W(a,b,x) denote ^(-^i h 2, -bx\ shew that the equation y= TF(a, 6, a?) is equivalent to a?-« Tr(6, a, y). 5. Shew that a second solution of the differential equation for F (a, 6, c, s) is a;»-«F(a-<J+l, 6-c+l, 2-c, x). 6. Shew that the equation («2+V)^+(«i-»-V)^+K-»-Wy=o can, by change of variables, be brought to the form and that this latter equation can be derived from the hypergeometrio equation by the substitution 6^^, x— , where m is infinitely large. 264 TRANSCENDENTAL FUNCTIONa. [chap. XI. 7. Shew that o:(z)^p / -1 00 1 I — « where C («) is the coefficient of A*» in the expansion of (1 - 2A«+A*)-«' in ascending powers of A. 8. Shew that, for values of x between and 1, the solution of the equation where il, ^, are arbitrary constants and F{af /3, y, x) represents the hypergeometric series. (Cambridge Mathematical Tripos, Part I, 1896.) 9. Shew that the differential equation for the associated Legendre function Pi!!*^{z) of order n and degree m is satisfied by the three functions ( 00 \ 1 2 p \ « W* ""^ — w 1 1-2 2^ ^" v-s"^ n+1 -^m 00 P\ n ~2 n-hl V 2 2 «-(««-l)* V, — m n+1 / 00 1 P\ n m - 1 2 2 " l3^ n+1 9n 1 V"T" "2 2 10. Shew that the hypergeometric equation (Olbricht) ^(^-i)S-{y-(«-»-i3+i)^}2+flft^-o is satisfied by the two integrals / f^-Vl-«)T^-^-^(l-a?«)-*(i« J and //-*<- f)— >{l-(l-«)f}— A. MISC. EXS.] HYPBRGEOMETRIC FUNCTIONS. 265 11. If (l-ar)*-^-T'i?'(2a, 2/3, 2y, ^) = l+^ar+Cx«-|-2>a?8-|-...,• 8hew that F{a, ft y+i, a?)/'(y-a, y-ft y+i, x) ''^+y+i^^y+i)(y+t)^^ ^ (y+i)(y+i)(y+|)^'^^ •- (Cayley.) 12. Prove that P^ («)=! tannir {§, «- Q.^.i («)}, where Pm(2) and Qn(^) cure the Legendre functions of the first and second kind of order n. 13. If a function F{a^ /S, /S', y ; x^ y) be defined by the equation F{a, ft /y, y ; x, y)° r(o)rty-„) /,*«""' (1 -«)'^~' (1 - «*)"* (I -^yT^du, then shew that between /* and any three of its eight contiguous functions F{a±l\ F{fi±\\ F{^±1\ F(y±l), there exists a homogeneous linear equation, whose coefficients are polynomials in x and y. (Leyayasseur.) 14. If y — a - 3 < 0, shew that, for values of x nearly equal to unity, and that if y— a— ^aO, the corresponding approximate formula is JPf a^ ^\ r(a+g) , 1 ^(o,fty,^)«- r(a)r03) ^^i^^- (Cambridge Mathematical Tripos, Part II, 1893.) 15. Shew that when \x\ < 1, /; ^^-'(v-ar)'*-*-!^— ^(l-v)-*'Ji' «--46'^sinair8in(p-a)ir. ^^^^°^,^^°^ i?^(a,ai,p,^X where c denotes a point on the finite line joining the points 0, x, the initial arguments of y-^ and of i^ are the same as that of ^, and that of (1 - v) reduces to zero at the origin. (Pochhammer.) CHAPTER XII. Bessel Functions. 146. Ths Bessel coefficients. In this chapter we shall consider a class of functions known as Bessel functions, which present many analogies with the Legendre functions con- sidered in Chapter X. As in the case of the Legendre functions, we shall first introduce the functions, or rather a certain set of them, as coefficients in an expansion. For all finite values of z, and all finite values of t except t » 0, the function can be expanded by Laurent's theorem (§ 43) in a series of ascending and descending powers of t If the coefficient of <^, where n is any positive or negative integer, be denoted by Jn {z), we have (by § 43) Jn{z)^^\yr---e^'^'"'^du, the integral being taken round any simple contour in the u-plane enclosing the point t^ = 0. To express this quantity Jn (z) as a power-series in z, write 2t z Thus '^'^(')-L{i)'h~^''*'^'^'' the integral being taken round any simple contour in the ^plane enclosing the point t^O. This can be written -.(')-^(l)M.^'®7'— '^ 146] BESSEL FUNCTIONS. 267 Now (§ 56) we have ^— . I ^"^'^Vd^sa the residue of the function ^-»--*^» e* at its pole, the origin. If n is a positive integer, this residue is 1 if n is a negative integer, say = — «, the residue is zero when r = 0, 1, 2, . . . « — 1, and when r ^ « it is 1 In any case, the residue is Thus if n is a positive integer, we have and if n is a negative integer, equal to -- 8, we have or Jn(z)=^('-iyJ.(z). Whether n be a positive or negative integer, the expansion can clearly be written in the form ~ ^^ rto 2'^*-r I r (n + r + 1) • The function Jn{z) thus defined for integral values of n is called the Bessel coefficient of the nth order. We shall see subsequently (§ 149) that the Bessel coefficients are a particular case of a more extended class of functions known as Bessd fuTtctions, Bessel coefficients were introduced by Bessei in 1824 in his " Untersuchung des Theils der planetarischen.Stdrungen, welcher aus der Bewegung der Sonne entsteht." In reading some of the earlier papers on the subject, it is to be remembered that the notation has changed, what was formerly denoted by Jn {z) being now denoted by J^ (2«). EaiompU 1. Prove that if (l-2a^-^)H46«^ ^■*" "^^ "^^ ••• ' then will ««8in6«=^i/i(«)+^2/,(«)+^s/j(«)+.... (Cambridge Mathematical Tripos, Part I, 1896.) 268 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. For replacing the Bessel functions in the given series by their values as definite integrals, we have ^(tR 2iriy / 2a 1 Y 46» ^ the integrals being taken round any simple contour in the u-plane enclosing the origin. Taking a new variable t, defined by the equation vre thus have ^.7i(z)+il^,« + ^j/,W + ...-2^.j" <«+6« ' where the int^;ration is now to be taken in the clockwise direction round any large simple contour in the ^plane. This expression is (§ 56) equal to minus the sum of the residues of the function at its poles t^ib and t=-ih ; that is, it is equal to 2t 2i or 0" sin bzj which is the required result Example 2. Shew that, when n is an integer, mss— 00 We have J^^^l^'D ^^i''^ Ji^^ , or 2 <"/,(* +y)= 2 <"/«(«) s rJr(y). »— — 00 ms-oo ra— 00 Equating coefEicients of t^ on both sides of this equation, we have the required result. 147. BeaseVa differential equation. We have seen that, for all integer values of n, the Bessel coefficient of order n is expressed by the formula where C is a simple contour in the ^-plane enclosing the point ^ « 0. 147, 148] BESSEL FUNCTIONS. 269 We shall now shew that the function Jn{z) is a solution of a certain linear differential equation of the second order, namely, For we find in performing the differentiations that -7r-7ik \ t-^''^e «1- 27rt2~Jc I n4-l ^ t "^4^ dt = 0, t-^. since the function e ** f^"^ resumes its original value after the point t has described the contour in question. Thus Jn {z) satisfies the differential equation d^Jniz)^ldJn(z) dz' dz + ^i-gj„(.) = o. • This is called BesseVs equation of order n. Its properties in many respects resemble those of Legendre's differential equation, which is also a linear differential equation of the second order. 148. BesseVs equation as a case of the hyper geometric equation. If c be any finite quantity, the differential equation of the hypergeometric function / n 00 tc T^-^'^c z ) — 71 — IC 2 ^ *^ ) is (§ 137) dz^ z dz \^z z — c J z(z — c) If in this equation we make c tend to an infinitely large value, we obtain 270 TRANSCENDENTAL FUNCTIONS. [chap. XII. which is Bessel's equation of order n. Thus BeswVs eqwUion ca/n be regarded as a limiting case of the hypergeometric equation, corresponding to the function / X c \ Limit P { n 1 . ic 2 "^ *^ ^ 0sac \ — n — ic 2 "■ *^ Another representation of BesseVs equation as a limiting case of the hypergeometric equation is the following. If we change the dependent variable in Bessel's equation, by writing e^u, the differential equation for u is easily found to be y= dz I /^. 1\ du fi n*\ ^ Now if c be any quantity, the differential equation of the hypergeometric function P\ n 00 1 2 8 — n I — 2tc 2ic — 1 IS dhi d^ ^ /I I 2 - 2ic\ du ^ /n^c ^ 3 ^\ u ^^ \z z — c J dz \ z 8 ) ziz — c) If in this equation we make c tend to infinity, we obtain cPu /I . o .\ du [ n« i\ ^ which is the above equation. Hence Bessel's equation is a limiting case of the hypergeometric equation, being the equation for the function e^ Limit P . n oc 1 2 8 — w T — ^io 2tc — 1 4 Bessel's equation is connected not merely with the general hypergeometric equation, but with that special form of it which we have considered in con- nexion with the Legendre functions. 148] BESSEL FUNCTIONS. 271 For the differential equation of the associated L^eudre function (§ 129) is (§ 138) the equation of the function P^ -1 X m - 2 " + 1 m I 2 m 2 m 1- _ 2n« or (§ 139) / 4n« 00 < \ m 2 m 1 n + 1 ^ -^ m 2 — n — m 2 The differential equation of this function is *y / 1 1. \ c?y / m» __ n + 1 m|\ nh/ i{z^yW -4ffi^ zV d{z^)\ z^-4m,^ n z^J 2^{z^'-An^) If in this equation we make n tend to infinity, it becomes =0. d{z^y^ 2^d{z ^l dy or r)-(->-?)|.=». which is Bessels equation. Thus Bessels equation of order m is the same as the equation for the function Limit PrT (l - 2nV' By considering Bessel's equation as a limiting case of the hypergeometric equation, we can deduce certain solutions in the form of definite integrals. For the differential equation of the function '^ 00 e ic is satisfied by the integral '(-:-y7.'-'(-0 ^\»+i-w« {t-zy^-hdiy 272 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. if C is a ooDtour such that after describing C the integrand returns to its initial value. When c becomes infinite, this expression reduces to sf^tr*^ j <-»-i(<-«)-»-ie«'cfc, which accordingly satisfies BesseFs equation if C be a contour of the kind described ; C can for instance be a figure-of-eight contour encircling the points t=^0 and t=:z. In fact, if we write we have =0. Other solutions can be found by changing the signs of n and t. Example. Shew that Bessel's differential equation is the limiting case of the equation of the hypergeometric function 00 c* P\ in i(c-n) ^ when c tends to infinity. 149. The general solution of BesseVs equation by Bessel functions whose order is not necessaHly an integer. We now proceed, in the same way as in § 116, to extend our definition of the function «/n(^) to the general case in which n is not an integer. It appears from the proof given in § 147 that, whatever n may be, the differential equation is satisfied by an integral of the form y=z'' r'*-^ e *^ dt, J c provided the path of integration C is a contour on the ^-plane, so chosen that the function e «r^-i resumes its initial value after describing C. 149] BESSEL FUNCTIONS. 278 Now when the real part of Ms a very large negative number, the function is infinitesimal. Hence y will be a solution of the differential equation, provided the contour G begins and ends with values of t whose real part is infinitely large and negative. Let therefore a contour C be taken which begins at the negative end of the real axis, and after proceeding close to the real axis to the neighbour- hood of the origin makes a circuit of the origin and returns, close to the real axis, to the negative end of the real axis again. The integral y taken round this contour satisfies BesseVs differential equation. We shall now shew that this solution y can be expressed in the form of a series of powers of z. Suppose as usual that by tr^^^ is understood that branch of the function ir^-^ which when continued (§ 41) to the point ^ = 1 by a straight path, arrives at the point ^ = 1 with the value unity. Then we have y = £«| r^>e^6 **<fo 00 / ly-.sr-fn r r=o ^.r\ Jc But (§ 100) we have But when n is an integer, we have (§ 146) ^"^^^^ ^to2^^r! r(n + r + l)* Comparing these results, we have, when n is an integer, •^" <^> = 2^- (l)7c'"^'*'"^'"' where C is the contour already described. Now we have seen that the right-hand side of this equation has a meaning and satisfies BesseFs differential equation for all values of z and all values of n ; whereas, up to the present, Jn{z) has been defined only for integral values w. A. 18 274 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. of n. We shall take this opportunity of extending the definition of Jn (z), in the following way. For aU valuea of n and of z, the function h 'Silr''"'"' CO (,l)r^gir+n rro2~+*-r! r(w + r-!-l) or will he denoted by Jn (z). In the integral, tr^"^ is to have the value which becomes unity when the variable t travels in a straight line to the point ^ = 1 : and is a contour which encircles the point ^ = and begins and ends at the negative end of the real axis in the ^plane. The function Jn(z) thus defined is called the Bessel function of z, of the first kind and of the order n ; it satisfies BesseVs differential equation of order n. Since Bessel's differential equation is unaltered by the change of n into — n, we see that J^n(z) is also a solution of the equation ; and therefore the general solution of BesseVs equation is of the form aJn (z) + bJ^n (z\ where a and b are arbitrary constants, except in the case in which Jn (z) and «/Ln (z) are not independent functions ; this exceptional case happens when n is an integer, for then, as we have already seen, we have the relation Jn (Z) = (- irJ-n (*). A second solution of BesseFs equation in the case when n is an integer will be given later. 160. The recurrence-formulae for the Bessel functions. As the Bessel functions, like the Legendre functions, are members of the general class of hypergeometric functions, it is to be expected that recurrence-formulae will exist between them, corresponding to the relations between contiguous hypergeometric functions (§ 145). We shall now establish these recurrence-relations ; the proof given does not assume the order n to be an integer, and consequently the formulae are valid for all values of n, real or complex. Let C be the contour described in the last article, which begins and ends at the negative end of the real axis in the ^plane, and encircles the point t = 0. Then since the function e ^ir^ ¥mmt^mBsm 150, 151] BESSBL FUNCTIONS. 275 is infinitesimal at the extremities of this contour, we have the equation 2n or */n-i (^) + */n+i (-8^) = — t/n (-2) (A), z which is the first of the recurrence-formulae. Next we have, by differentiation, or ''-^.- -,■''<.')- J-^M (B). From (A) and (B) it is easy to derive other recurrence-formulae, e.g. ^-i^ = \[j^,{z)-j^^,{z)] (C), and ^!^) = j^,(,)_^/,(,) (D). Example 1. Shew that 16'^^=^.-4(«)-4^«-,W+6/»(«)-4y,^,(*)+J,^,W. Example 2. Shew that 161. Relation between two Bessel fimctions whose orders differ by an integer. The various recurrence-formulae found in the last article can* however be easily deduced from a single equation, which connects any two Bessel functions whose orders differ by an integer, namely n+r w^/ ly _*; (-ly z"^^ "" ^ (zdzy where n is any number (real or complex) and r is any positive integer. 18—2 276 TRANSCEKDENTAL FUNCTIONS. To establish this result, we have, by § 149» dr {Jn{z) d{s^y\ z"" dr I f d{j^y27n.2''Jc e *tdt dt [chap. XII. (- 2/ -2^+'" •^ n+r V-*^)* which is the equation required. The recurrence- formulae can be derived without difficulty from this resultw Thus, equation (B) of the last article is obtained by taking r = 1 in this equation : and equation (A) of the last article may be derived in the following way. Taking r= 1 and r = 2 successively in the formula just proved, we can express the first and second derivates of Jn(z) in lerms o{ Jn{z), Jn+i{z) and Jn+2(z), in the form dJn (z) _ n J. ,. J , . — ^^ =- - (1 - n)Jn (Z) y-^n+i W + «/n+s(4 Substituting these values in BesseVs equation we have Changing n to (n — 1) in this result, we have J,+,(z)-^/,(*)+J._,(^) = 0, z which is the formula (A) of the last ai*ticle. The other recurrence-formulae can be derived in a similar way. 152, 153] BESSEL rUNCTIONS. 277 162. The roots of Besael fwncHons, The relation established in the preceding article enables us to deduce the interesting theorem that between any two consecutive recU roots of Jn {z) there lies one and only one root of Jn+i (^)*. For since Jn(z) satisfies BesseFs equation, it follows that the function y = Jir^ Jn (z) satisfies the differential equation From this equation it is evident that if f be a value of z (real and not zero) for which ^ is zero, then the signs of -~ and y must be unlike at the point 2r = f . Now let -^ = ^i and ^ = f, be two consecutive roots of the function -^. It is clear from the differential equation that neither y nor -~ can be zero at either of these points. Then the function -^ -~ has a different sign just before reaching 2^ = fa to that which it has just after leaving -? = fi; and hence it follows that the function y -^ has a different sign just before reaching z= ^^to that which it has just after leaving z^ fj. The function y must therefore have an odd number of roots between the points 2r = f 1 and z = fj. But from Rollers Theorem it follows that y cannot be zero more than once in this interval : so y must have one and only one zero between the points 2^ = f 1 and 2^ = fa • and therefore the zeros of y and of -p occur alternately. Thus, between any two consecutive roots of the function z~^ Jn (z) there lies one and only one root of the function -j- {z~^ Jn (z)] or - z~^ J^^i (z): which establishes the theorem. 163. Expression of the Bessel coefficients as trigonometric integrals. We shall next obtain a form for the Bessel coefficients (ie. the Bessel functions for which the order n is an integer), which in some respects corresponds to the Laplacian integrals obtained in ^ 119 and 132 for the Legendre functions. * The proof here given is dae to Oegenbaaer, MonaUhefU fiir Math, ym. (1897). 278 TRANSCENDENTAL FDNCTIONS. [CHAP, XII. If Id the equation we write t = e^, we have n»-ao Changing i to — i in this equation, we have »« -00 Adding and subtracting these results, we have 00 COS {z sin ^) = 2 J^ (z) cos 7uf>, n« - sin (z sin <^) = S Jn (z) sin n^. n«-flo Since J^ (z) = (— l)'*«7Ln (A these equations give cos(* sin <l>)^Jo (z) + 2J, (^r) cos 2^ + 2J4 (2r)c08 4<^ + ..., sin {z sin ^) = 2Ji (^) sin ^ + 2/, (5) sin 3^ + ... . As these are Fourier series, we have (§ 82) J^ (-?) = - I cos nO cos (2fsin 0) dO, (n even), 1 r* = - I cos ntf cos (z sin tf ) dtf , (n odd), 1 f* J^ (fr) = - I sin 72^ ain (2? sin 0) dd, (n odd), 1 r* = ~ I sin nd sin (z sin ^) dtf , (n even). TT Jo Since cos (ntf — e sin tf ) = cos nO cos (^ sin 0) + sin ntf sin (z sin tf), we have in all cases when n is an integer 1 f* Jn (z)=i— I COS {nd — J? sin 0) d0, TT J Q the formula required. Example, To shew that for all values of n, real or oomplez, tbe integral 1 fw y—- I ooe (n$ - $ mn $) dS ir J 154] BESSEL FUNCTIONS. 279 satisfies the differential equation ^ ^ dy . f-, w'N sin nn /I n\ which reduces to BessePs equation when n is an integer. 1 /■» Forif y = - I cos(n^-28ind)eW, wehave 7^^~ I sindsin (?i^-«sin^)cW, so cPv 1 r* -^= I sin*^cos(n^-«8in^)cW, p y+^ = - / cos*dcos(7id-«sind)cW, and " "J"^ ^y^~ \ sin(nd— 2Sind)cW — I pCos(nd-«smd)cW. Now integrating by parts, we have -I — — sin(n^-2sin^*e^«— sinnTT-l — I — (n-;^cos^)cos(w^-«sind)eW, vjo z nz vjo z ^ and therefore 1 t A -- ^ «= — sinn7r--«- / cos(n^-«sind).rf(n^-«sind) nz «"«• y ^»o 1 . n . sin?tir /l _»\ which is the required result. 164. Extension of the integral-formula to the case in which n is not a/n integer. We shall now shew how the result 1 f » (xr) = - I cos {nd — zsmff) dO must be generalised in order to meet the case in which n is not an integer, ie. the case of the Bessel functions^ as opposed to the Bessel coefficients. 280 TRANSCENDENTAL FUNCTIONS. [CHAP. XII, Suppose that the real part of z is positive. Write ^ = ^ ru in the formula we thus have where u"*^^ has that value which becomes unity when the variable u travels by a rectilinear path to the point w = 1. Since values of t whose real part is large and negative correspond to values of u whose real part is large and negative, we see that the path in the u-plane, along which this integral is to be taken, is still a path leading from u = — x round the point u = and returning to w = — oo . Let this contour be chosen so as to consist of (a) a straight line parallel to, and below, but indeBnitely close to, the real axis from it = — oo to w = - 1 ; ()8) a circle I of radius unity described round the origin ; (7) a straight line parallel to, and above, but indefinitely close to, the real axis from u = — 1 tot4 = — 00. Thus '^»<^> = 2^/_">-^'^^""-^'^« + 2^/,«-""^ 1 /•-• ?U-^) ZTrt j-i where u~*"* has in the first integral the value «<*■♦■»)*» at u = — 1, and in the third integral has the value 6~<'^*^*' at u = — 1. Hence, writing w = — t in the first and third integrals, and u = e^ in the second integral, we have 1 r* ^(»+i)t» /•• -^-f+-\ dt where, in the last two integrals, t"**~* has the value 1 at the point t«l. Writing t = «•, we have ^ sin(n + l)7r r^_^.,^„h#^^ TT Jo ^trjo ' TT Jo 154] BES8EL FUNCTIONS. 281 or Jn(z) = - ['coaizsine-neyde-''^^^'^ I e-»»-"inh«d^ (1). This formula is valid when the real part of £: is positive. When the real part of z is negative, a similar procedure leads to the result J^ (z) = ^— j r cos (z sin + n^) d0 - sin nir j e-«*+'»inh<> ^^1 (2). When n is an integer, the formula (1) gives 1 f* J„(^) = - j cos (w^ — 2: sin ^) d^, 'jr Jo when the real part of -^ is positive ; and the formula (2) gives J„ {z) = tllZ! r cos {ne + z sin 0) d0, or, since /« i^) = (- 1 )** J-^ {z\ 1 f"" t/n (-2^) = - cos (nd — 2^ sin 0) d0, TT Jo when the real part of z is negative. Thus in either case when n is an integer, we have again the result of the last article, namely the formula '„(^)=1 rcos(/i^-2rsin^)(i^ (3). The equation (3) was kDown to Beasel. Equation (1) is due to Schlafli, Math. Ann. m. <1871) ; equation (2) was first given by Sonine, Math. Ann. xvl (1880). The trigonometric integral-formula for J^ (z) may be regarded as corresponding to the Laplacian definite integrals for the Legendre functions. For we have seen that the Bessel function J„^ (z) satisfies the differential equation of the function Limit P^ 11—00 or or But the Laplacian integral shews that this quantity is a multiple of ^* I'ob "£"' "^ (0 " £')*" i}*««*J*=°"'^ '^ Limit I fl-i — cos<^] ooHfmf>d<l>f I ^OM^GOBin<f>cUl>j the similarity of which to the above result (3) will be observed. 282 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. 166. A second expression of Jn {s) as a definite integral whose path of integration is real. Another definite-integral formula, which is valid for all values of z and a certain range of values of n, can be obtained in the following way. The function Jn (z) is expressed for all values of n and z by the series I (-l )r^+«r rro2'»-^r!r(n+r+l)* Since (§ 95) we have this can be written in the form (-l)-z"^r(r + l) Jniz)= S ■0 2»r(n + r+l).2r!rQ) * Now by § 107 we have provided the real parts of (r + ^ ) ai^d (n -f = ) are positive. Thus if the real part of ( '^ + «) ^ positive, we have •^nW«-— TXr— 7 n 2 ^ — ^ cos^'V^sin'^i^d^. 2«ryjr(w + ijr=o ^r\ Jo Tj ^ / .X ^ (-l)''^cos*'^ But cos (£^ cos <p) = 5) ^— -. , ^ . ^ ^' r^^Q 2r! Thus we have . J (z) =s J, I J .1 I COS (2: COS 6) sin** ^ (i<5. 2-rQ)r(n+y'o This formula is true for all values of z, and for all values of n whose real part is greater than - ^ . Example 1. Shew that P^ (cos 6) « jT^^fYj j^ e''<^» Jo (^ ffli^ ^) ^ dx. 80 155, 156] BESSEL FUNCTIONS. 283 For we have fry Jo ir J J = r(n+l)P»(co8^), which establishes the result. Example 2. Shew that Pn"» (cos ^)= -7—4-7 fx f e''^9J„,(xamB)x^dx. r(n— m+i;y o (Cambridge Mathematical Tripos, Part 11^ 1893.) 166. Hankers definite-integral solution of BesseVs differential equation. If in the result of the last article we write t = cos (f>, we obtain the result 2"r(i)r(n-.y^-i It will now be shewn that this integral is a member of a very general class of definite integrals which satisfy Bessel's differential equation, namely, integrals of the form y=^n f e^ (t^ ^ l)n'h dt, Jo where C may be any one of a number of contours in the ^-plane. The importance of solutions of this type was first shewn by Hankel*. To shew that integrals of this class satisfy BessePs equation, we form the first and second derivates of the expression y, and find that = -z^' ( {^e«'((»- !)"+»- (2n + l)He^ («•- I)""*} J c dt * Math. Ann. i. 284 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. From this it is clear that Bessels equation will be satisfied by the integral J c provided is a closed contour such that the integrand resumes its initial value after making a circuit of C The similarity of this result to the general theorem of § 142 is very apparent. 167. Expression of Jn (z), for all vaiues of n and z, by an integral of HankeVs type. We shall now shew how the particular solution Jn {z) of Bessel's equation can be expressed by an integral of HankeFs type. Consider the contour formed by a figure-of-eight in the ^plane, enclosing the point ^ = + 1 in one loop and the point ^ = — 1 in the other, so that a description of the contour in the positive sense involves a turn in the positive direction round the point < = + 1 and a turn in the negative direction round the point ^ = — 1. After turning round the point < = + 1 in the positive sense, the integrand resumes its original value multiplied by ^»-*>*'*, as can be seen by writing it in the form ««+(n-i)log«-l)+(i»-4)log(e+l). ^ » and after turning round ^ = - 1 in the negative sense, it is further multi- plied by -(i»-i)fcrt Hence after describing the whole contour, the integrand resumes its original value. rd+.-i-) Thus y^i^\ e**(e«-i)»-*(ie is a solution of the differential equation, valid for all values of z and of n ; the symbol (1 +, — 1 — ) placed at the upper limit of the integral indicating that the path of integration consists of a positive revolution round 1 and a negative revolution round — 1 In this equation we shall suppose as usual that ^ has the value which reduces to I when z travels by a straight path to the point z^\, and we shall suppose (^" — I)*"* to have initially the value which reduces to «-<*-*)'» when t travels by a straight path to the point ^ = 0. To find the relation between this quantity y and the particular solution Jn {z) of Bessel's equation, we expand y in the form 157] BESSEL FUNCTIONS. 285 To evaluate the iutegrals which occur in this series, write F{r,n)^j (1+.-1-) m+,-1-) Then F(r, n+ l) = j (r+«-r)(<«- !)»-*(£« a+.-i-) ^H-i -I /•(i+.-i-)(««_l)«.+i(r + l)^- „, . Thus we have F{r, n) = - ^ ^"T" jP(r, n + 1). This result enables us to reduce the evaluation of F(r, n) to the evalua- tion of F(r, n + 1), and thus to the evaluation of F(r, n + k), where A is a positive integer so chosen that the real part of (n + A:) is greater than — ^ • We have therefore to evaluate the integral F(r, n)=|' where we may now suppose that the real part of n is greater than - ^ . The contour can be supposed to start at the point t = 0, where (^'— I)**"* has the value «-<'»-*>»», then to proceed to the neighbourhood of the point ^ = 1 along the real axis, then to make a positive turn in a small circle round t = 1, then to return along the real axis to the point ^ = 0, where (^*— 1)**"* has now the value e<»-*)'^, then to proceed along the real axis to the neighbourhood of the point ^ = — 1, then to make a negative turn in a small circle round ^ = — 1, and lastly to return along the real axis to the point t = 0, where (^— l)**"* has now the value e"^""*^**. Since the real part of n is greater than — 2 , the integrals round the small circles at ^ = 1 and ^ = — 1 are infioitesimal, and we therefore have ^0 Jo dt JO Jo where in each of these integrals the quantity (1 — <■)*"* is now supposed to 286 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. have the value unity at ^ = 0. Writing — ^ for ^ in the two last integrals, we have F(r, n)= - {e^"-*)** - «-<»-*)'<} {1 - (- 1)^-1} JV(1 - t*)^-^dt = — 2% cos* -x-sin(n— sJtt/ v* <^"*^ (1 — i/)*-* dv, where v = <", <by § 105) = - 2i C08' ^ sin (n - 1) ^fi (^ , n + ^) (by §106) =-2ico8'^sin(n-l)^ ) r \^' ' r^n+2 + l) This result has now been proved to hold so long as the real part of n is greater than — = : and in virtue of the formula F(r. n) g^^^ Fir.n + 1), we see that it holds universally. Thus we have F(r, n) = 0, when r is odd ; and it is therefore sufficient to take r even. Let r = 2s. Then the formula becomes ^•(2,, n) = 2i sin (» + 1) ^ \^/; \ , V . But (§ 97) we have r(„+l)rQ-„) — TT sm Therefore 2^V^/'« + l^ /'(2», n)=— ^^ -_i U . r(i-n)r(n + , + l) and so y = 2 ^^ — a-, M < '"> 2«! r (i-n)r(n + «+l) or ,^ » (-l)^+» ^^"^^ (§) ^ .=« 2-«! r(i-n)r(» + ,+ i)- But "^ ' ,ro2-+»«!r(n + «+l)* 158] B£SSBL FUNCTIONS. 287 Therefore J„(.)=_^-^,. (ly This formula gives the required expression of J^ {z\ It is valid for all values of n and of z\ but when n is of the form (^ + 2)» where A is a positive integer, the factor T (o — w ) becomes infinite and the integral / becomes zero (since the integrand is now regular at all points within the contour), so that for this exceptional case the formula is indeterminate. Example, Deduce the formula from the result of this article. « 168. Bessel functions as a limiting case of Legendre functions. We have already (§148) shewn that Bessel's differential equation of order 771 is the same as the differential equation of the associated Legendre functions We shall now express this connexion more precisely, by establishing the formula ^„ (.)= Limit n-.P,«.(l-^.). For taking the expression of the associated Legendre function by a definite integral (§ 131), we have ~mpm(-t _ ^ (n-\-m){n + m-\) (n-TO + 1)^" / ^N** " " V 2»»y" (2to - 1) (2m - 3) l.-n-.n"" \ 4m*} /„'{^-£« + '^*(-| + £')P«^""^'''^' and as n becomes infinitely great, the right-hand side of this equation tends to the limiting value 7^5 rr-To ^r = / ( 1 + - COS ^) sin"*Ad<f>, (2m — l)(2m — 3)...l .TrJo \ n v ^ ^' 288 TRANSCENDENTAL FUNCTIONS. [CHAP. XU. — o)... 1 . TT Jo (27?i-l)(2m-3) or (§165) Jn.{zy> which establishes the result stated ; it is due to Heine*. 169. Bessel functions whose order is half an odd integer. The result of § 157 suggests that when the order n of a Bessel function Jn (z) is a number of the form A? + 5 , where A? is a positive integer, certain exceptional circumstances arisef in connexion with the function. In this case it is in fact possible to express the Bessel function •^*^^^^"2*+*r()fc + ?) I 2(2A: + 3)'*'2.4.(2A; + 3)(2A + 5) ••*) in terms of well-known elementary functions. For by § 161 we have, if A; be a positive integer, /„«-(-2)..«4^|-'.W}. But the series-expansion of the function J^ {z) is 2\i . sin^r. . , , 2M f, z* z* ) / 2 \ Therefore J,^i {z) -^ ^ ^^,^ (-^J , which is the required expression of the function Jj^ {z) in terms of more elementary functions. The student will without difficulty be able to prove that a second solution of Bessel's differential equation in this case is * Heine's definition of the associated Legendre function is somewhat different from that which has since become general and which is adopted in this book : this leads to differences of statement in many other formulae, such as that of this article. t The student who is familiar with the theory of linear differential equations will observe that in this case, and also in the other exceptional case of n an integer, the difference of the roots of the '* indicial equation *' of BessePs equation is an integer. 159, 160] BESSBL FUNCTIONS. 28& Example. Shew that the solution of the equation IS y-« 4 2 Cp{*^-m-* (2apl*)+»J'« + i(2a^)}, p»0 where the quantities Cp are arbitrary constants, and o^y a|, ... o^mi &x^ ^^ roots of the equation o***^— i (LommeL) 160. Expression of Jn (z) in a form which furnishes an approximate value to Jn {z)for large real positive volumes of z. We now proceed to form an integral which will be found to play the same part in the theory of the function J^ {z) as the integral of § 104 plays in the theory of the function T {z). We shall suppose 5 to be real and positive. Then, by § 155, we have, for all positive values of n, J^ (z) SB / cos {z cos <f>) siD^<f> cUf>, Writing cos <^ = a?, this becomes Jn(z)=^ J 11 I (1 - a^y^ COS zxdx, or Jn(-^) = Real part of /^ . /^(l - x')'^e^dx. In order to transform this integral, we take in the plane of a complex variable t a contour OPQBCOy formed in the following way. is the origin (^ = 0) ; P is the point ^ = 1 — />, where /> is a small quantity, and OP is the part of the real axis between and P. Q is the point ^ = 1 + 1/), and PQ is a quadrant of a circle which has its centre at the point ^ = 1. B ia the point t^l -hik, where A; is a large positive quantity, and QB is the line (parallel to the imaginary axis in the ^plane) joining Q and B, C is the point t = ik, and BC is the line (pcurallel to the real axis) joining B and C, Lastly, CO is the part of the imaginary axis between C and Q. Then the function is regular at all points of the ^plane in the interior of the contour OPQBCO ; and therefore the integral taken round this contour, is zero. w. A. 19 290 TRANSCENDENTAL FUNCTIONS. [CHAP. XIL We can write this relation in the form f +f +f +f +f =0. J OP J PQ JOB JbC J CO Now the part of the integral due to PQ tends to zero with p, and the part due to BC tends to zero as k becomes infinitely great, while the part due to CO is purely imaginary. Thus we have Real part of / = — Real part of I , J OP J QB and so Jn W= Real part of , ^ ,i f (1 -fy*^^cU. In this integral write %u z 80 that u varies between the limits and oo when t describes the line QB; and (i_^)* = /f(?y«*(l + gy. and therefore f =2"-*w {"*'""^r-»-*rr^M-* (l + g)*~*dM. Thus we have or j; («) = r(n+i).(27r^)» '-{-(-i)i}/>»-i(>4r-('-ir}*'' -"{'-(«4)f}/:— {(>-ir-('-£n<<'«j This is the integral-expression required. It is easily seen to furnish an approximate value of J^ (z) for large positive values of ^ ; for as ;? becomes indefinitely large, the two integrals in the expression tend respectively to the limits 2r (n + ^ j and zero ; and therefore the function Jn (z) approximates for large positive values of z to the value (l;)*^|'-KI)l}- 160] BESSEL FUNCTIONS. 291 The evaluation of «/» (z) when z is large will be considered in fuller detail in the next article. The result of this article can also be obtained in the following quite different manner, which connects it more closely with the general theory. We have seen in § 148 that Bessel's differential equation is a limiting case of the general hypergeometric equation, represented by the function 00 e n i-2tc 2w?-l Since the differential equation of the P-fimction (0 00 c a p y Z is (§ 142) satisfied by the integral taken between suitable limits, we see that Bessel's equation is satisfied by the expression Limit «<•«-* (<»-» (l-^y •-»+«• («-0*'*<^, or ««•«-* [ r-J «-«< {z - <)•"* dt, or (putting < «= - ivz) «<•«-» jif''-hs^'k{z+%vz)*-he-^zdv, or «*V» / (v+tv«)*-*e->"cfo. The limits of the integral can be taken to be and oo , since these satisfy the conditions for the limits found in § 142 ; and hence it follows that is a solution of Beesel's equation. Similarly the quantity is a solution of Bessel's equation. The solution J^ {z) must therefore be of the form J^(z)»A^i^ r {v+iv^'ie'^dv'¥Be'*'s^ riv-iv^'^'ie'^dv, 19—2 292 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. where A and B are constants independent of i. This is substantially the form given above, but the determination of the constants A and B \b a, matter of some difficulty, for which the student is referred to a memoir by Schafheitlin, CrelU^s Journal^ oxiv. p. 39. Example, Shew (by making the substitution u^2zcot<t>m the integral found above^ or otherwise) that ir J^{s)^ ?^^^^?!^. fV««~**oos«-*<^oosec*»+i<^cos{«-.(n-i)^}c&^. r(n+i).v*Jo 161. The Asymptotic Expansion of the Bessel functions. The Bessel functions can for large values of the argument be represented by asymptotic expansions. We shall here consider only the asymptotic expansion of Jn{^) for positive real values of z; this was discovered by Poisson (for n = 0) and Jacobi (for general integer values of n). The theorem has been considered for complex values of z by EUmkel * and several subse- quent writers. We shall derive the asymptotic expansion from the integral-expression Jn(z) = (iwz)^ r (n + 1) ■-('-(- 1)1} /:--'i('4r-(' -sn^» found in the last article. It is first necessary to find the asymptotic expansion of the integral which we shall denote by the symbol /. Now we have iu ^.jt2Li^)r(g..)-(>,.^.^. * Math, Ann, i. 161] BESSEL FUNCTIONS. 293 Therefore tu ^ Mt-i)^..(t-.) jv^.._/;(g-.)-(i^.)^.^, or where Now as z becomes infinitely large, n having any definite finite integer value, the remainder-term R^^ tends to the limit It follows from this that Limit z^R^ = 0, and therefore the series r(ifc+ 1) jlH- i^(A^ + r)(^-Hr-l)...(fc-r + l)0| is the asymptotic expansion of the function r e^u^fl + ^^j'du (k>0). Substituting this result in the expression already found for «/,(-?), we see that rU^) (2,r^)* r(n + i) ^r(" + 2)2tf + i. W (fe^l ^- i f ■ l\-^)f^ (""^•^'')("~^+'')-("~^-^5)i'-^.-,-.(-t)r | COS 294 TRANSCENDENTAL FUNCTIONS. [CHAP. XIL is tbe asymptotic expansion of the Bessel function «/n(^) for large positive values of z. Even when z is not very large, the value of c/» {z) can be computed with great accuracy from this formula. Thus for all values of z greater than 8, the first three terms of this asymptotic expansion give the value of Jq (z) and Ji (z) correct to six places of decimals. 162. The second solution ofBesseVa equation when the order is an integer. We have seen in § 149 that when the order n of Bessel's differential equation is not an integer, the general solution of tbe equation is where a and fi are arbitrary constants. When however n is an integer, we have seen that and consequently the two solutions Jn (*) and «/_n (z) are not really distinct. We therefore require in this case to find another particular solution of the differential equation, distinct from Jn(^)t in order to have the general solution. To obtain this second solution, we write y = uJn (z), where u is a new dependent variable, in Bessel's equation Remembering that Jn(^) is a solution of Bessel's equation, the differ- ential equation for u becomes 162] BESSEL FUNCTIONS. 296 dj^ a dz . 1 rt or -_ + 2-y— — + -=rO. dz Integrating this equation, we have UtUt log ;j- + 2 log Jn {z) -h log z = constant, dvt 6 . « • or -=- = . - . .. , where 6 is a constant, dz z [Jn {z)Y where a and 6 are arbitrary constants. The complete solution of Bessel's equation can therefore be written in the form To find the nature of the solution thus obtained, we observe that in the vicinity of the point ^ = the integrand is of the form ^~**~* (constant + powers of f)~*, which when n is a positive integer can be expanded as a Laurent series in the form The function l'tr'[Jn{t)]''dt has therefore the form where the quantities d^^, ^^-m+s* ••• ^^^ definite constants. It thus appears that the complete solution of fiessel's equation can be written in the form y = ilJn(^) + 5{/»(^)log^ + t;}, where v is the result obtained by multipl3nng together J^{z) and a Laurent series of the form 296 TRANSCENDENTAL FUNCTIONS. [CHAP. XIL where the quantities (£-an> <2-sn+af ••• ^^^ definite constants, and A and B are arbitrary constants. The expansion of Jn (^) being known, we see that the product V has the form jt^ X a power-series in ii* ; and thus a second solution of BesseVs differential eqiwiion, in the case in which n is an integer, can be taken of the form Jn (z) log 2r + 2r« (Oo + OiZ* + Oi-e* + a,;^ -h ...), where the quantities o^, Oi, a,, ... are definite constants. These quantities a are not however all of them strictly speaking definite, since by adding a multiple o{ Jn{z) (which will leave the expression still a solution of Bessel's equation), it is possible to change all the quantities a after On-i. This solution will be denoted by Kn{z)*. The coefficients Oq, Oi, a,, ... may theoretically be determined by substi- tuting this expansion in the differential equation, and equating to zero the coefficients of successive powers of z, A better method is however the followingf. We have seen that when n is a positive integer, J.^(z) reduces to (— l)^Jn{z); in fact, if in the equation T ( \ l^X'"'^^ T (-1)^ /^\^ J-(«-.,W=^2^ p:or(-n + 6+p-hl)r(2> + l)V2; we suppose the quantity € to tend to zero, all the terms of the series vanish as far as p = n, since r(— w-hp + l) is for these terms infinite. Changing the meaning of the index of summation /> in the other terms, we have ./_(„_, w W p.«r(-n+e+;)+i)r(p+i)Uy ^ ' \v ,tor(€+p + i)r(n+p + i)V2y ' and when e = 0, the first of these partial series is zero and the second is (-l)»J„(e). Since the quantity (-1)»J_,^,(^)-J,„_,(^) vanishes with e, we can take as a second solution of BesseVs equation the limiting value of the quotient * In referring to memoirs it most be borne in mind that different writers have taken different definitions of the Besael fonctions of the second kind, t Dae to Hankel, Math, Ann, x. p. 470 (1869). 162] BESSEL FUKCnONS. 297 Substituting the above values for the Bessel functions, this becomes \2) ^toT(p + l)€T{-n + e + p+l)\2) '*'V2/'pto^ ^ e \2) ' where /(€) represents the expression ^^^^"r(n+p + i)r(p + 6+i)(2J "r(n+p-€+i)r(p+i)V2J ' The limiting value of /(€)/e, as € tends to zero, is ^ 21og^ ^ ^<^ + ^> 1 r(«+p+i) r(p+i){r(n+p +!)}•• Also, since we have limit ,r(_„^^,^^^i) = (- l)"^' r (« - p). Consequently we obtain, as a second particular solution of Bessel's equation, the expression* \2; ptor(p+i)V2; ^{2) pZ,,r(n+p+i)T{p+i)\2) z r(n+p + l) F(p + 1)] r"'*2 r(n + p + i) r(p + i)j* The coefficient of log« in this expression is 2J,(«). So, dividing the expression by 2, we have tbe second solution in the form ' j.(«)i»g.-^;i;E^(i)''-^.wiog2 + (iXi. (-i)p cy^l ^(n+p+ i) r(p+i )] W pr«r(n+p + l)r(j) + l)V2/ 2 1 T(n+p + l) r(p + l)j It is convenient to add to this e^cpression a term /,(.){log2+M)|. * This 18 Bankers seoond solution Y^ (z). It is really dn ^ ' dn ' 298 TRANSCENDENTAL FUNCTIONS. [CHAP. XU. which is itself a solution of Bessel's equation ; so the second solution now takes the form ^n(.)iog.-2^2— |j-^y ^\i^S\ (- \y /^\^ [F (n -h p + 1) r (p +1) 2r (i) ] 2Wptor(n+p+i)r(i)+i)V2>' lr(^+i) + iK r(p+i) r(i) r This is the solution K^ {z) which we take as our standard. Since, when r is a positive integer, we have r(r + l) F(l)_ 11 1 r(r+l) r(l) '■^2'^3'^"'^r' we can write K^{z) in the form _|(|ri^z^j,+ui+...+i+i+|+...+ i_i(|r. 2\2/ ^«o (»H-l>)lpl I 2 3 p 2 7i+pJ V2/ When n is an integer, the two independent solutions of Bessel's differ- ential equation are J^ {z) and K^ (z\ Example 1. Shew that the function K^ (z) satisfies the recurrence-formulae These are the same as the recurrence-formulae satisfied by J^ («). Example 2. When the real part of 2 is positive, shew that the expression r8in(isin<^-n<^)ci(^- j e-*«toh«{e««+(-l)*e-«^ dS is a second solution of Bessel's differential equation of integer order n. (SchlaflL) Example 3. Shew that the expression yologi+2(y,-^4+iJ5-...) is a second solution of the Bessel equation of order zero. 163] BESSEL FUNCTIONS. 299 163. Nexmianns expansion ; determination of the coefficients. We shall now consider* the expansion of an arbitrary function f(z), regular at the origin, in a series of Bessel functions, in the form f(z) = OqJo (z) + a^Ji (z) + aa/2 {^) + " •» where the coefficients a©, «!, ««> ••• are independent o{ z. , Suppose first that such an expansion is possible, and let us try to determine the coefficients, by expanding both sides of the equation as power-series in z and equating coefficients of the several powers of z. Since /(^)=/(0)+2(|)/'(0) + |(|)V"(0)+|^(|)V"'(0) + ... and «^»W = ^(|) {l-i!(„Vi)(|) +2!(n + lKn + 2)(l) --}' we have on comparing coefficients the equalities /(O) = «.. 2/' (0) = a„ 2"/" (0) = - 2a. + Oa, etc., from \rhich without difficulty we find a. = /(O). «„ = 2 1/(0) + JV" (0) + "'^"4;"^'^ /'^ (0) + . . . + 2»-»/ <«) (0)| (n even). «„ = 2|n/'(0)+"-(^^V"(0) + »(»'-l')("'-3') ^,T) (0) + . . . + 2»-'/ <») (0)1 (n odd). These coefficients take a simpler form, if we introduce functions Oi {z), Ot (e), 0, (z), ..., defined by the formulae ^,. » n(n*-V) n (n» - !•) (n» - 3") 2»-in! . ,,, for then it is easily seen that On is twice the residue of the function On(t)f(t) * 0. Nemnann, Theorie der Bet$eV$ehen Functionen. The exposition here given followB Eftpteyn, AmuUet de Vkeolt Normate (8) z. p. 106 (1898). 300 TRANSCENDENTAL FUNCTIONS. [CHAP. XH. at the point < =» 0. The two formulae for On (^) can be united by reversing the order of the terms ; thus OnW«-^jrH-|l + 2(2n-2)"^2.4(2n-2)(2n-4)"*"-J' the series terminating with the term in z^ or jf^K We thus have Neumann's expansion f(z) = Oo/o W + «i/i (z) + a, J, (z) + ..., where ao=/(0)/ and On (n > 0) is twice the residue of 0» {t)f{t) at the point ^ = 0, so that On^^^l On(t)f{t)dt, y where 7 is any simple contour surrounding the origin. 164. Proof of Neumanria expansion. The method by which this result has been found cannot be regarded as a proof, since the possibility of the expansion was assumed. We can, however, now furnish a proof by determining directly the sum of the series obtained. From the definition of On {z\ we can at once obtain the identities 0„+. {z) + 2 ^^ - 0„_, {z) = 0, (n > 0). d tV ".W'-s©. Writing the first of these equations in the symbolic form On+x-2DOn-On.i-0, where i)-^, and solving the series of recurrence-equations obtained by giving n integer values, in the same way as if D were an algebraic quantity, we obtain for On the symbolic expression On(^) = i[{-i) + (i>+l)»}-+{-i)-(i>'+im(j)- This symbolic expression can be transformed into a definite integral in the following way. 164] BESSEL FUNCTIONa 801 1 r* We have - - / e~*^ du, t Jo where the upper limit must be understood to mean that direction at infinity which makes the real part of tu positive and infinite ; and therefore or, writing tu = a?, On (t) = f *i tr^' e'^[[w + (^ + ^«)*}« + {x^(a^ + <»)*}«] dw, Jo where the upper limit now means the real positive infinity, so that the integration may be regarded as taken along the real axis of x. Writing this in the form <'-<«-r?rj:[r-i^T^<-'>-{^T(^/]-'^' we have 0, (<) J, («) + 2 2 On (<) J,, («) (by § 146) « ^ Limit I 2^ < * xH»^+pM . e"* dx ^t x«oo Jo y Limit / t JT.flo Jo ^ — -« 6* " cia?. In order that this integral may have a meaning, the real part of —r— V must be negative, a condition which is fulfilled when If this inequality is satisfied, we have therefore Oo{t)Jo(z) + 2 X On{t)Jniz) = :r^ . « From this result Neumann's expansion can at once be derived ; for let /(z) be any function which is regulai* in the interior of a circle C whose centre is at the origin, and let ^ be a point on the circumference of the circle. Then if z be any point in the interior of the circle, the condition | jp | < | ^ | is satisfied, and therefore we have ^ ^Oo(t)Jo(z) + 2iOn(t)Jn(z). t-Z n«l 302 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. Thus /(^) = ^— . y^— •^ ^ 2'fnJc t — z = 2^|^/(e)d^ jOo (0 Jo (z) + 2X0n{t) Jn (^)} where flo =/(0) and On = — . f On (0/(0 ^^ (^ > <>). This establishes the validity of NeumaDn's expansion for points z within the circle C, Example, Shew that cos f-^o W- 2/, (i)+2/4 («)-..., 166. Schldmilch*8 expansion of an arbitrary fwnction in terms of Bessel functions of order zero. Schlomilch * has given an expansion of a quite different character to that of Neumann. His result may be stated thus : Any function f(z) which is finite and continuous for real values of z between the limits z=^0 and z^ir, both indu^sive, muy be expressed in the form f(z)^ao + Oi Jo (-8^) + Oi Jo (2-er) + a^Jo (3-8^) + . . ., where Oo = / (0) + - f ' w f \l - ^)-*/' (ut) dt du, ^Jo do an^- I ucosnu I (1 - ^*)~*/' (ut) dtdu (n > 0). ^Jo Jo Schl5milch's proof is substantially as follows Suppose that F and /are two functions connected by the relation f(z) = lf\l-^)-iF(zs)ds. Then we have 2 p f(z) = -[\l-'^)-^sr(zs)ds. TTJo ♦ ZeiUehnftflT Math. u. Phynh, n. (1S67). 165] BESSEL FUNCTIONa 803 In this equation, write zt for z, multiply both sides by -er (1 — ^«)-* dt, and integrate with respect to t between the limits ^ = and t^\. Thus ^ ['(1 - t^yif(zt)dt - — (\l - «•)-* dt l\l - l^)'^8F'(Z8t)d8 Jo IT J Jo 2 r» /•(*•-»•)* where x^zst^ y^z8{l — t^)^. Performing the integrations, we have -/ {s^-a^''fy^F'{x)dxdy, TT J Jo f \l - e«)*/' (^) (ft = F{z) - J?'(0). Jo 2: , 'o Now by the definition of the function/, we have /(0)^F(0). Thus F{z) =/(0) + z f\l - <«)-*/' (zt) dt. Jo This equation expresses the function F explicitly in terms of the function /, whereas in the original definition / was expressed explicitly in terms o{ F. In order to obtain Schlomilch's expansion, it is merely necessary to apply Fourier's theorem to the function F(zs). We thus have f(z)^- ( (l-«»)-*(fojl ('F(u)du + - 1 rcosnwcosn^m)^^! TT^o {jrJo ^ i»«lJo j If' 2 * T' = - I F(u)du + — 2 / cos nuF{u) Jo (nz)diu ^Jo ^ »»i Jo In this equation, replace F{u) by its value in terms oi f{v). Thus we have + - i J, (tl?) r cos nw 1/(0) + u f (1 -«•)-*/' (we) (ftl dw, ^ »-i Jo I Jo ) which is Schlomilch's expansion. Example, Shew that if < f < ir, the expression |*-2 |^,(*) +5^o(3»)+i^.(6*)+...} 804 TRANSCENDENTAL FUNCnON& [OHAP. XIL is equal to z ; but that, if ir ^f ^ 2ir, its value is «+2irC08-i--2(*«-ir*), where cos'^ - is taken between and -x . z 3 Find the value of the expression when » lies between 2«r and Zw. (Cambridge Mathematical Tripos.) 166. TainJ^iXtion of the Bessel functions. Many numerical tables of the Bessel functions have been published. Meissel's tables (Berlin, 1889) give the functions Jo(z) and Ji{z) to 12 decimal places for real values of z from jer = to er = 15^, at intervals of 001. Tables of the second solution F© (z), defined by the equation z^ I 1\ ^ Y,{z)^J^{z)\o%Z'\r-^^''\\ + 27 2rT« ■*■•••' from ^ s= to ir = 10*2, are given by B. A. Smith, Messenger of Math, xxvL (1897). The British Association Reports for 1889, 1893, 1896, contain tables of the functions /n(^)> which are solutions of the differential equation dhb \du f. ^n^ ^ so that In(^)^i'^Jn(i^)' A table of the first 40 roots of Jo (^) is given by Wilson and Peirce, BtM, Amer. Math. Soc. in. (1897). Miscellaneous Examples. 1. Shew (ag. by multiplying the expansions for e^ ~<^ and «~^ " if , and equating the terms independent of t) that {^oWl'+S {^1 (*)}»+2 {y, «}»+2 {J, (»)}»+... = !, and hence that, for real values of s, J^ {z) can never exceed unity, and the other Bessel coefficients of higher order can never exceed 2~l. 2. Shew that, for all values of fi and v, MISC. EXS.] BES8EL FUNCTIONS. 305 3. Shew that 4. Shew that «^nW w+1- n-h2- w+3-../ 5. Shew that /_mW«/m-iW+«^-m+iW*^mW== 2sin/i9r «-« 6. If v^-/! be denoted by Q^ («), shew that (Lommel.) dz z 7. Shew that ^_%ii).l_i(!LhI)e.w+,{e.W}«. 8. If the function — I 2^co8'^t«cos(mu~;;sint«)(;?ti JJ{z)= 2_ :^,(i«)'»^-m.*.p, (which when k is zero reduces to a Bessel Amotion) be denoted by J^* (z\ shew that where N^m, *. p is the " Cauch/s number ** defined by the equation Shew further that this function satisfies the equations and ^m'^\z)=2mJ^' (z)-^2(k+l){jLi{z)-j!^+i{z)}, (Bourlet) 9. If quantities v and i/'are connected by the equations Af^E-eainEy cosi;=_ jz. where |6|<1, «hewthat v=jr-h2(l-fl8)i 2 i (i«)»/^*(m«)-sinmJ/; -where J^*(^)=- I (2costi)*cos(mti-;?8ini«)(;?t^ "■y 10. Prove that P,-(cos^)=^^/^|(a.*+y«)i ^1^, •where a«rcos^, ^+y*»»r*sin* $, and c„*» is a numerical quantity. (Cambridge Mathematical Tripos, Part II, 189a) W. A. 20 306 TRANSCENDENTAL FUNCTIONS. [CHAP. XII. 11. Shew that, if n is a positive integer and (m+2n+ 1) is positive, (Cambridge Mathematical Tripos, Part I, 1899.) 12. Prove that 2 r* Jo (2)=- I sin (« cosh «) fl?«. (Cambridge Mathematical Tripos, Part II, 1893.) 13. Prove that and if Y^ (z) is HankeFs second solution of Bessel's equation, defined by the equation i r, (.) = Limit '^-.(^)-'^«(^)<=<^^ , T »=faitefer sm nir ■^••^ ?'--»-i=-5i?<Mi,('*£r'(^')- 14. Shew how to express ^J^ (z) in the form AJ^ z)-{-BJo{z\ where A, B are polynomials in z ; and prove that J4(6*)+3^o(6*)-0, 3./fl(30*) + 6^,(30*)=0. (Cambridge Mathematical Tripos, Part II, 1896.) 16. Prove that, if •/„ (a{)=0 and J^ 03ft=O, ^^^xJ^{QX)M^x)dx^O, and |^*^{*^H(aa.0}«d:r-if8{^^^^(^)j8. Hence prove that the roots of J^{x)=0^ other than zero, are all real and unequal. (Cambridge Mathematical Tripos, Part I, 1893.> 16. Shew that /* ^V~2~/ a:-*»+"»J«(ar)(ir=2-*+'«a'»-"»-i— --^— ^— , if 2w + l>f/i^-l. (Cambridge Mathematical Tripos, Part I, 1898.). MISC. EXS.] BESSEL FUNCTIONS. 307 17. Shew that !- i '^{^^w n pao 2 ^"ICiJ (Lommel.) 18. Shew that the solution of the differential equation where ^ and ^ are arbitrary functions of «, is y-(^y<^-^^(^)+^-^-'W}- 19. Shew that ^^y /^V^(«sin^)sin'*+i^cW«2-*y«+j(«). 20. In the equation the quantity n is real ; shew that a solution is given by ( - 1)"» z*^ cos {u^-n log z) (Hobson.) cos (n log «) - S «-i 2»*m!(l+n2)*(4+n2)* (wi^+n^)* * where t^^ denotes tan-i?+tan-i5+...+tan-i-. 12 m (Cambridge Mathematical Tripos, Part II, 1694.) 21. Prove that the complete primitive of the differential equation where m is a positive integer, is u^AI^{z)+BK^{z\ where, for real values of z^ ^" ^=1-73^^1) /." *-«-'"*«'>l>'"*'^- Prove also that /eo (««+««)-'»-Jco8Md«. 20—2 308 TRANSCENDENTAL FUNCTIONS. [CHAP. Xn. Shew that for very small values of z, jro(;^)--log|--677..., and that for very large values of z, (Cambridge Mathematical Tripoe, Part II, 1898.) 22. If C be any ciu*ve in the complex domain, and m and n are integers, shew that j^O^{z)0^(z)dz^O, where k^O \f the curve does not include the origin ; and, if the curve does include the origin, it=0 if m + w, its=2»rt if m=n. CHAPTER XIIL Applications to the Equations of Mathematical Physics. 167. Introduction : illustration of the general method. The functions which have been introduced in the three preceding chapters are of very great importance in the applications of mathematics to physical investigations. Such applications are outside the province of this book ; but most of them depend essentially on one underlying circumstance, namely that by means of these functions it is possible to construct series which satisfy certain partial differential equations, known as the partial differential equations of mathematical physics; and in this chapter it is proposed to explain and illustrate this fundamental property. The general method may be explained by considering first the solution of the partial differential equation ^■^dj^=^ <1>' a solution which, while resting on the same principles as those to be developed later, does not require the use of any but the elementary functions of analysis. Consider any solution V{x, y) of this equation (1). Near any point at which a branch of the function V{Xy y) is a regular function ctf x and y, and which we may without loss of generality take as origin of coordinates, this branch of the function V{x, y) can by Taylor's Theorem be expanded as a power-series of the form ^(^» y) = ao+ Oifl;-!- 6iy -h o,^ -h 6a^ 4- Cay* + Os^:* + ; on substituting this value of V in equation (1), and equating to zero the coeflScients of the various powers of x and y, we obtain the relations Oa + c, = 0, 3a8 + Cs = 0, Sdj + 6, = 0, 310 TRANSCENDENTAL FUNCTIONS. ' [CHAP. XIII. Fixing our attention on those terms in V which are homogeneous of the nth degree in x and y combined, it is clear that the equalities just written will furnish (n — 1) relations between the (n + 1) coefficients of these terms of degree n. When these equations are satisfied, there will therefore remain only {(n + 1) — (ti — 1)} or 2 coefficients really arbitrary in the terms of the nth degree in F. Now the expressions and F=(ir — iy)* satisfy equation (1), and therefore if A^ and B^ ate any arbitrary constants, the expression An {x + lyY ^Bn{x- iyY satisfies equation (1), and is homogeneous of the nth degree in x and y, and contains two arbitrary constants. It therefore represents the most general form of the terms of the nth degree in V\ and so the general solution of equation (1), regular at the origin, can be expressed in the form F(a:,y) = ilo + ^(a? + ty) + 5i(a?-iy) + il,(^ + iy)« + A(«-iy)"+ (2), where the quantities Aq, A^, B^, A^, ... are arbitrary constants. This expansion furnishes the general solution of equation (1) ; what is however in general needed is the particular solution of equation (1) which satisfies some further conditions. As an example of the conditions most fi'equently occurring, we shall suppose that the value of the required solution V(x, y) is known at every point of the circumference of a circle, whose centre is at the origin and whose radius is any quantity a ; it being supposed that this circle lies wholly within the region for which V is regular. This being given, we shall shew that the constants Aq, Ai, B^, ... can be found, and the solution can be completely determined. For writing x = r cos 0y y^r sin 0, the value of V is known when r = a, as a function of 0, say f(0). Let the function /(^) be expanded as a Fourier series in the form f(0) = Oo + Oi cos ^-h 6i8in ^ + Oacos 2^ + 63 sin 20 + (3), where the coefficients a©, Oi, 61, a,, ... are given by the formulae 1 r*' \ •2ir 1 [^ an = - I /(t) COS ntdt IT J Q 1 f^' 6n = — / f(t) sin ntdt y (*). 168] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 311 Consider now the expression ao + ^(oi COS ^ + 61 sin 0) + Q (a, cos 2^ + b^ sin 20) + (5). This expression (0) reduces to (3), i.e. to /(^), when r=^a\ and since we have r^cosn0=^ {(x + iy)* + (a? - iy)»}, r« sin w^ = 2^ {(a? + iy)« - (a? - iy)**}, it is clear that the expression (5) is of the form (2), i.e. that it is a solution of the equation (1). It follows that the solution V of equation (1), which is characterised by the condition that it has the value V=f(0) when r = a, is given by the expansion where r fr\^ F= Oo + -(aicos ^ -h 61 sin ^) 4- -) (o^ cos 20 + 6, sin 2^) + ..., a \af I 1 f^' < 1 f*' a% = - I f(t) cos ntdt, 1 f^ bn = - I f{t) sin ntdt \ ^ Jo The principal object of this chapter will be to obtain theorems analogous to this for the other partial differential equations of mathematical physics ; the method followed will be in most respects similar to that by which this result has been obtained. 168. Laplace's equation; the general solution; certain particular solutions. The partial differential equation a»F d^V d'V ^ da^ dy^ dz^ is known as Laplace's equation, or the potential-equation, and is of importance in the investigations of mathematical physics. The general solution of this equation was given by the author in 1902. It may be written r2ir F= I f(x cos t-^-y sin t + iz, t) dt, Jo 312 TRANSCENDENTAL FUNCTIONS. [CHAP. Xni. where / is any arbitrary function of the two arguments x cos t + y sin t + iz and t The solution is eflFected in Monthly Notices of the Royal Astron. Soc, VoL LXii. In this chapter however we are -concerned not so much with the general solution as with the particular solutions which satisfy certain further conditions. To the consideration of these we shall now proceed. Let the equation be transformed by taking instead of the independent variables a?, y, z, a new set of independent variables r, 0, ^, connected with them by the relations x^rsmO cos ^, y = r sin ^ sin ^, z = r cos 0. It is found without difficulty* that Laplace's equation becomes drV dr)^ siu^e a<^« "^sin 0d0\^ dO)"^' Let us seek for particular solutions of this equation, of the form V^Re<P, where R, 0, <I>, are functions respectively of r alone, alone, and ^ alone. Substituting, we obtain 1 d f.dR\ 1 d ( , ^de\ 1 d^ Rdr\ dr)^(bHm0dd T^ ^ dff) "*" cDsm*^ d<^« ""' Now the quantity Rdr\ drj does not involve ^ or ^ ; and since by this equation it is equal to 1 d / . ^d0\ 1 d^ @8in0d0V^ dd) 4>sin^^d<^«* it clearly cannot vary with r : it is therefore independent of ?•, ^, and <^, and so must be a constant ; this constant we shall write in the form n{n'ir 1). ^,(^f)-»(.-i)^-o. We thus have dr Write r ^ 6**, so dr = e^du. Then this equation becomes d / ., dR^ ^U''£)-^<^^'^''=' d^R dR , , 1 \ z> A or -FT +-3 n(n+l)R = 0. du^ du ^ * The ^ork is given in full in Edwards' Differential Calctdut, 168] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAI. PHYSICS. 313 This is a linear diflferential equation of the second order with constant coefficients ; its solution, found in the usual way, is where A and B are arbitrary constants. The most general form of the function R is therefore Considering next the function <I>, it can in the same way be shewn that the quantity is independent of r, 0, and (f>, and so must be a constant. Writing this constant in the form — m', we have for the determination of <I> the equation ~— 4- m*4> = of which the general solution is <I> = a cos m<f> + h sin m<^, where a and h are arbitrary constants. It thus appears that the expressions r"cosm<^0 and r*sin77i^© are particular solutions of Laplace's equation, if n and m are any constants and is a function (of 6 only) which satisfies the equation ^ ^ %%\n0dd\ ddj sin*^ Writing cos^ = 2r, this becomes © = 0. But when m is a positive integer, this is (§ 129) the equation which is satisfied by the associated Legendre functions of order n and degree m ; so a particular solution is the function Pn'^iz), or P„«»(cos^). Hence generally we see that the (2n+ 1) expressions r'^Pn (cos ^), r'* cos <^ Pn^cos ^), r~ cos 20 Pn' (cos ^), ..., r'^C0S7K/>Pn'*(C0S^), r" sin <^ Pn"^ (cos 6), r^ sin 20 Pn^ (cos 0), ..., r^ sin iKf> Pn^ (cos 0), where n is a positive integer, are particular solutions of Laplace's equation. 314 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII. Moreover, since PfJ^ (cos 0) is of the form sin"* ^ x a polynomial of degree (n — m) in cos 0, it is easily seen that each of these quantities, if expressed in terms of x, y, z, becomes a polynomial, homogeneous of degree n, in a?, y, z. It can in fact be easily shewn, by using the result of § 132, that r" cos m<f> Pn^ (cos 0) is a constant multiple of f J Q •2ip (a? cos t + y sin. t-h iz)^ cos mt dt, and that r^ sin m<l> Pn^ (cos 0) is a constant multiple of /, Sir (x cos t + y sin t-^ izY sin mt dt, from which their polynomial character is evident; these forms have the further advantage of exhibiting these particular solutions as cases of the general solution given at the beginning of this article. Example, If coordinates r, d, ^ are defined by the equations y = (r* - 1)* sin S cos (f), .2= (r*- 1)* sin $ sin <^, shew that the function 7= P^'» (r) P„»» (oos $) cos mit> is a solution of Laplace's equation 169. jf^c series-solution of Laplace* s equation. The particular solutions of Laplace's equation, which have been found in the preceding article, enable us to express the general solution, in the form of an infinite series involving Legendre functions. This series-solution will of course be really equivalent to an expansion of the general solution /, Jir fix COS ^ H- y sin ^ + u, t) dt already mentioned ; but the series-form is (as will appear from § 170) more convenient in determining solutions which satisfy given boundary-conditions. For let V(Xj y, z) be any solution of Laplace's equation da^ "^ dy^ "*■ dz^ " 170] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 315 Then in the neighbourhood of any ordinary point, which we may take as the origin of coordinates, V can be expanded in the form Substituting this expansion in Laplace's equation, and equating to zero the coeflScients of the various powers of x, y, z, we obtain an infinite number of linear relations between the coeflScients a©, Oi, 61, Ci, Oj, .... There are ^n{n'-\) relations of this kind between the ^(71+ l)(n-h2) coeflScients of terms of degi'ee n in the expansion of V: and so only -{2 (n + 1) (n + 2) — s w (n — 1)1 or (2n + 1) of the coeflScients of terms of degree n in the expansion of V are really independent. But in the last article we have found (2n + 1) independent polynomials of degree n in a?, y, z, which satisfy Laplace's equation, namely the quantities r^Pn (cos 6\ r~ cos <\>Pf^ (cos 0), , r~ cos n<f>Pn^ (cos 0), r^Bin (f>Pn^(cm0)y , r^ sin TK^Pn** (cos ^). It follows that the terms which are of degree n in a?, y, z in the expansion* of V must be a linear combination of these (2n + 1) quantities ; that is, V must be expansible in the form F = ilo + r [A^P^ (cos 0) + ^1^ cos ^P,^ (cos 0) + B,^ sin <I>P^^ (cos 0)] + r« {A^P^ (cos 0) + Ai^ cos <^P,» (cos 0) + A^* cos 20 P,' (cos 0) + Ba^ sin i^Pa^ (cos ^) + J8,« sin 20 P,^ (cos ^)} + . . . , where the quantities A^, A^, A-^y B^y ... are arbitrary constants. 170. Determination of a solution of Laplace's equation which satisfies given boundary conditions. In order to determine the unknown constants Aq, Ai^ A^y B^y ..., which appear in the expansion just found, it is necessary to know the remaining conditions which the function V is required to satisfy. A condition of frequent occurrence is that V is to have certain assigned values at the points of the surface of a sphere, which we may take as being of radius a and having its centre at the origin. This sphere will be supposed to lie entirely within the region for which F is a regular function of its argtiments a?, y, z. When r = ay F is therefore to be equal to a given function f{0y <f>) of and 0. The constants Aq, Aiy -4l^ JS/, ..., are therefore to be determined fix)m the equation f{0y ^)^Ao + a {ill Pi (cos 0) + ili» cos (f> Pi^ (cos 0) + A' sin <l>Pi^ (cos 0)] + a* {A^Pi (cos 0) + -4,^ cos (fyP^^ (cos 0)+ ...] + ... 316 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII. In order to obtain the value of one of these constants, say -4n'*, from this equation, we multiply both sides of the equation by Pn** (cos ^) cos m^, and integrate over the surface of the sphere. On the left-hand side we thus have r i '/(^» ^) -Pn*" (cos d) COS nKf> sin dd d(f>. Jo Jo As to the right-hand side, we know that /, 2ip is zero except when r = m, and that /•2» COS m<f> cos r<f> d<l> I COS m<f> sin r<f> dif> Jo is always zero ; and also (by § 130) that f ' Pr"^ (cos 6) Pn"" (cos 0) siu d0 Jo is zero except when r = n. It follows that on the right-hand side, every term vanishes except the term a'^^n"* re {Pn"* (cos 0)Y cos» m</> sin d0 d<f>. Jo Jo Since I cos' m^ d<f> = tt, and (by § 1 30) /J {P„» (cos 0)}^ sin 6 d0 = ^^^ <^| . this term has the value 2n-|- 1 (n — m)! We have therefore the formula ^""^ = i^»' • (^! /o' ly^^- '^^ ^-^ <^°« ^> ""^^ "^<^ «^ ^ '^^ '^*' which determines the coeflScients -4n"* in the expansion of V, The coeflScients £„"* can be similarly determined : and so finally the solution V of Laplace's eqxuition, which has the value f{0, <f>) at the surfa^ of the sphere, is given for points in the interior of the sphere by the expansion ^ = Jo ^^ ^ (a)7«' ly^^'- *^'^ {^" ^"^^ ^^ ^" ^*'*'' ^^ + 2 2 J^^^^;P„'»(co8 5')-Pn"(cosd)cosm(^-^')|8in^d^d<^'. 171] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 317 This result may be regarded as a three-dimensional analogue of the two- dimensional result of § 167. Elxample 1. Shew, by applying the expansion-theorem just given, that P^ {cos ^ COS ^ + sin ^ sin ^ COB (<^ - <t/)} — P„ (cos 6) P„ (cos ff) + 2 J^ |J^; P,- (cos e) P,« (cos ff) COS m (* - <t>'). Example 2. Prove that if the product of a homogeneous polynomial of d^ree n in 0?, y, z and the function P„{cos^cos^+sin^sin^cos(^-^')} be integrated over the smface of the sphere, the result is 4ir/(2n+l) multiplied by the value of the polynomial at the point (^, ^'). (This can be proved by taking ^ to be zero, which involves no real loss of generality, and expanding the polynomial by the theorem of this article.) 171. Particular solutions of Laplace's equation which depend on Bessel functions. It is possible to construct solutions of Laplace's equations in series in several ways, of which that which has been given, and which depends on Legendre functions, may be taken as representative. A fall discussion of the other methods would be beyond the scope of this book, but a general idea of them may be inferred from the result which will next be established, namely that the Bessel functions furnish a group of particular solutions of Laplace's equation, just as the Legendre functions do. When Laplace's equation da^ dy^ dz* is expressed in terms of the " cylindrical coordinates *' z, p, ^, where p and ^ are defined by the equations \x^ p cos <^, [y = p sin ^, it takes the form dz^ dp" '^ pdp "^ p^d<l>^ Let us seek for particular solutions of this equation, of the form F = ZP<I), where Z, P, <I>, are functions of z alone, p alone, and ^ alone, respectively. On substituting this value of F, Laplace's equation becomes ld«Z l/d|P ldP\ 1 d«<l> Zdz*"^ ?\dp''^ pdp)'^ p^d<f>^'' I 318 TRANSCENDENTAL FUNCTIONS. [CHAP. Xin. This equation shews that the quantity Z dz^ must be a constant independent of z, p, and ^ ; let this constant be denoted by i^. Then on solving the equation we have the particular solutions Z = e** and Z—e"^. Similarly the quantity 1 d^ <t>dif>^ is a constant, which may be denoted by — m' ; on solving the equation we obtain the particular solutions <I> = cosm<^ and <I> = 8inm^. The equation to determine P is now >-Jfn'- $)--«■ dp On putting kp = y, this becomes BesseUs equation of order m, *''+i^+ii -?")'-»■ df y dy a particular solution of which is P = /m(y). It follows that the expressions e"^ cos 7n<f>Jm{lcp) and ^±** sin m^t/in (A;p), where k and m are arbitrary constants, are particular solutions of LapUice's equation, 172. Solution of the equation — + ~ + 7=0. We now proceed to consider another partial differential equation. 173] APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 319 We have seen in the last article that Laplace's equation da^ dy'* dz* is satisfied by the particular solutions ^Jn(r) cos 710 and e* /„ (r) sin w^, where a? = rcos^, y = r8in^. But if we write TT^eT, where F is a function of x and y only, the Laplace's equation for W becomes It follow^ that, for all values of n, the quantities Jn (r) cos nO and /» (^) sin nO are particular solutions of this latter equation.. From these particular solutions, as in the case of the solution of the equation already described, we can build up the general solution of the equation d^V 3'F — + 4- F=0 00 in the form V= ^ J^ (r) {an cos nO + bn sin nO), n=»o where ao, Oi, a,, ..., 6i, b^, ..., are arbitrary constants. 173. Solution of the equation 3»F d*V a»F ^ ^ 1 4- — I- F = In order to solve the equation ^ a«F 3«F which is likewise of great importance in the investigations of mathematical physics, we first express the equation in terms of new independent vaiiables r, 0, (f>, defined by the equations a; = r sin ^ cos (f>, y=^r sin sin (f>, ,z =rcos^, 320 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII. and theD endeavour to find particular solutions of the form F=iee<i>, where -B, ©, <I>, are functions respectively of r alone, 6 alone, and <^ alone. Proceeding as in § 168, the diflFerential equation becomes 1 d/ d^N 1 d ( . ad^\ . 1 d^ ■*■ 22 rfr V dr j "^ sin ^ d^ V®*" df>; '•' 4> ain» ^ d^» ° "• This equation can be solved by the process used in § 168 for finding particular solutions of Laplace's equation ; the quantity ^ RdrK dr) must be a constant, which we shall denote by n (n + 1). If in the resulting «quation we write y = Rr^, it becomes (. . !)1 which is BesseUs equation of order (^ + s) • The quantity R can therefore be taken to be R^r^Jn^^{r), The equations for % and <I> are now found to be the same as those which occur (§ 168) in the solution of Laplace's equation; and proceeding as in § 169, we find that the general solution of the partial differential equation ^ 9>F a»j regular near the origin, can be expressed in Reform F= i r-iJn^{r) n=o AnPn (cos 0) -h An' COS <t>Pn' (cOS ^) + . . . + An"" COS n<l>Pn'' (cOS 0)] -I- Pn' sin <f>Pn' (cos ^) + . . . + 5„« siu TK^Pn** (cOS 0) ) + Bn' si] where the quantities A and B are arbitrary constants. 173] APPUCATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS. 321 When a particular solution V of the equation is to be determined by the condition that it is to take prescribed values at all points on the surface of a sphere, the constants A and B are determined exactly as in § 170. Example, Shew, as a case of the general expansion of this article, that »sO Note, The partial differential equations of §§ 172, 173, possess general solutions analogous to that of Laplace's equation. The solution of the equation of § 172 is where /is an arbitrary function ; and the solution of the equation of § 173 is where / is an arbitrary function. For the proof of these results, reference may be made to papers by the author. Miscellaneous Examples. 1. If a solution V of Laplace's equation be symmetrical with respect to the axis of i, and have the value V^f(z) at points on that axis, shew that its value at any other point of space is F-i f '/{«+» (^+y')* cos*} ^• tr Jo 2. Deduce from the result of Example 1 that the potential of a circular ring of mass M, whose equation is IS - f'if[c«+{«+i(a:»+y*)*co8<^}«]"*(i0. ^ J 3. Let P (x, y, ;r) be a point in space, and let the plane through P and the axis of z make an angle ^ with the plane ix. Let this plane cut the circle whose equations are in the points a and y, and let the angle aPy be denoted by $ and log (Pa/Py) by <r. If <rj Bf 4>'be regarded as coordinates defining the position of the point P, shew that Laplace's equation ^ a«F a«F takes the form d_ ( 8inh«r ?n . 1 [ sinh a 8 Fl 1 ^^o 9<r toosh«r-coe^ 8<rJ 8tf lco8h<r-cofl^ ^j sinh^ a (cosh a - cos $) d<f)^ ~ ' and that the quantities F— (cosh o--coa tf)* cos «^ cos WM^i^_, (cosh c) are solutions of it. W. A. 21 CHAPTER XIV. The Elliptic Function ff (z), 174 Introduction. li f{z) denote any one of the circular functions sin z^ cobz, tau ^ ... , it is well known that f(z + 2,r) =/(^), and hence that f{z + 2n7r) =/(^), where n is any positive or negative integer. This fact is generally expressed by the statement that the circtdar functions admit the period 2ir, They are on this account said to be periodic functions; and in contradistinction to other classes of periodic functions, which will be introduced subsequently, they are called singly-periodic functions. It will in fact be established in this chapter that a class of functions exists possessing the following properties : if f(z) be any function of the class, then f{z) is a one-valued function of z, with no singularities other than poles in the finite part of the ir-plane ; moreover, /(-?) satisfies, for all values of z, the equations /(* + 2a,.) =/(«), where cdi and ©j are two quantities independent of z. Functions f{z) of this class are said to admit the quantities 2(!i)i and 2o>2 as periods, and are called doubly-periodic Junctions or elliptic functions. The two periods 2(»i and 2a>s pl^y ^^^ same part in the theory of elliptic functions as is played by the single period 27r in the theory of circular functions. By repeated application of the formulae written above, we obtain as the characteristic equation of all elliptic functions the equation f{z + 2mft), + 2no)8) = / W» where m and n are any integers. 174, 175] THE ELLIPTIC FUNCTION ^{z). 323 176. Definition of |> {z). The elliptic functions may^ as we have just seen, be regarded as a generalisation of the circular functions. It is natural therefore to introduce them into analysis by some definition analogous to one of the definitions used in the theory of circular functions. One mode of developing the theory of the circular functions is to start from the infinite series 1 ** 1 z^ m^±i{z-miry' It can be shewn that this series converges absolutely and uniformly for air values of z except the values -8^ = 0, ±7r, ±27r, ±37r...; And that it admits the period 27r. If now its sum be denoted by (sin ^)~*, and this be regarded as the definition of the function sin z^ then from this definition we can derive all the properties of the function sin z^ and thus a complete theory of the circular functions can be developed. Similarly, as the basis of the theory of elliptic functions, we form the infinite series r^ + 2 {(^ - 2ma)i - ^nto^-^ - (2mft), -h 2n<»a)-«}, ^here a*i and o), are any two quantities, independent of z, whose ratio is not purely real, and where the summation extends over all integer and zero {except simultaneous zero) values of m and of n. It has been shewn in § 11 that this series is absolutely convergent for all values of z, except the values -^ = 0, ± Wi, ± Wj, + o), ± o),, ± 2©! ± ©a, ... . By comparing the series with the convergent series 2(tn* + n')~* as in •§11, it is seen that this convergence is also uniform (§52). The series therefore represent'S a one-valued function of z, regular for all values of the variable z except the values z = ^mo)^ + 2no>, ; and at these points, which are the singularities of the function, it clearly has poles of the second order. We shall denote this function by the symbol fp^z). Its introduction is due to Weierstrass. There are other ways of introducing both the circular and elliptic fiinctions into Analysis ; for the circular functions, the following may be mentioned : (1) The geometrical definition, according to which sin z is the ratio of one side to the hypotenuse, in a right-angled triangle of which one angle is z. This is the definition usually given in the introductory chapter of treatises on Trigonometry : but from our point of view it is defective, as it applies only to real values of z. (2) The definition by means of the infinite product sin«-z(l-5)(l-2gi)(l-^).... 21—2 324 TRANSCBNDBNTAL FUNCTIONS. [CHAP. XIV. (3) The definition by the inversion of a definite integral, We shall see subsequently that alternative definitions of the elliptic functions exist, analogous to each of these definitions (1), (2), (3), and that they may if desired be taken as fundamental in the theory. Example, Prove that ••<"-<'K£.)'.L~~^('-^')' 176. Periodicity t and other properties^ of |> {z). The function ^(z) is an even function of z^ Le. it satisfies the equation For if —z be substituted for z in the series which defines |> {z\ the resulting series is the same as the original series, except that the order of the terms is changed. But since the series is absolutely convergent, this change in order does not affect the value of the sum of the series; and therefore we have if>(^)=if>(-'^)- Further, the function |f> {z) admits the quantity 2o>i as a period. For = (2: + 2(k)i)-» - r-« 4- S {(^r + 2a)i - 2ma)i - 2no>^y^ - (^ - 2mo)i - 2n<»a)"'} = S {(2r - 2 (m - 1 ) Oh - 2na)8)-« -{z- ^nuo^ - ^am^)-^] . where the last summation is extended over all integer and zero values of m. and n without exception. But this last sum is zero, since its terms destroy each other in pairs. Thus we have Similarly f{z+ 2(k)a) = ^ {z\ and generally V{^ + imcoi -h 2rw«>g) = fp (z), where m and n are any integers. Therefore the function fp(z) admits the two periods 2ft)i and 2w2. Differentiating the above results, we see that jf>' (z) is an odd function of z, and admits the sams periods as ^ (z). 177. The period'paraUelograms, The study of elliptic functions is much facilitated by a method of geometrical representation which will now be explained. V^PP 176 — 178] THE ELUPnc function ^ (z). 325 Suppose that in the plane of the variable z we mark the points z^O, z s= 2(i(>x, z a 2(!i)9, z = 2a>i + 2a>s, ... and generally all the points comprised in the formula z = ^ramx + 2na>8, where m and n are any positive or negative integers or zero. By joining the point ^ = by a straight line to the point ^ = 2o>i, then joining the point 2o)i to the point 2(k>i + 2o)„ then joining the point 2o)i + 2(0, to the point 2a>s, and lastly joining the point 2a>9 to the point ^ = 0, we obtain a parallelogram in the ir-plane, which we shall call the fundamental period-parallelogram. It is clear that the whole ^-plane may be covered with a network of parallelograms, which are each similar and equal to this parallelogram, and which can be obtained by joining the other marked points by straight lines. These parallelograms will be called period-parallelograms. Then if < be any quantity, the points z^t, ^ = t+2a>i, 5«5t+2a>2, ..., £: =» ^ -h 2tna)i + 27Mi),, manifestly occupy corresponding positions in these parallelograms; these points are said to be coryruent to each other. It follows from the fundamental property of ^{z) that the fanctum f(z) has the same valvs at all points which are congruent unth each other ; and hence that the values which the function fp{z) has in any periodrparallelogram are a mere repetition of the values which the function has in any other period- parallelogram. 178. Expression of the function fp (z) by mea/ns of an integral. We shall now obtain a form for |> {z) in terms of an integral, which will be found to be of great importance in the theory of the function. The quantity (p {z) - r^, or 2 [{z — 2nK0i — 2na),)"* — (2mah -h 27k»,)~*}, is a regular fuuction of z in the neighbourhood of the point z^O, and is an even function of z. It can therefore by Taylor's theorem be expanded, for points z near the origin, in the form where clearly we shall have ^ = 32 (2mo>i -h 2no)a)-^, ^ = 52 (2mo>i + 2wo>a)^. Thus j>(^) = r-+g^ + g^ + .... 826 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV. Forming the square and the derivates of this expansion, we have |»"(«)-6*-^ + ^ + |^,*»+.... Therefore jf** («) — s if>"(^) " n^'* **" **™"8 involving z* at least. It follows that the function is regular in the neighbourhood of the point z^0\ and as it is doubly-periodic (for clearly any power or derivate of an elliptic function is likewise an elliptic function) it must be regular in the neighbourhood of each of the points z «s 2mcDi + 2n(k>s. But the only singularities of ^{z) are at these points : and therefore the only possible singularities of the function are at these points. The latter function is consequently regular for all values of z\ and so by Liouville's theorem (§ 47) is independent of z, and therefore is equal to the value which it has at the point z^O, which is j^^s* We have therefore the relation P'(^)=^ii»"(^) + r2^.. Multiplying by 3 jf>' {z) and integrating, we have where c is a constant ; on substituting the expansions in this equality, we find that c = ^g^. Thus, finally, the function p (z) satisfies the differential equation jp'» (^) = 4 jf>> (^) - 5r, jp (^) - 5r„ where g^ and g^ (called the invariants) are given in terms of the periods of jp(<j) by the equations ^r, = 60 2 (2ma)i + 2no),)~*, 5r, = 1402 (2?na)i + 2na),)"*. 178] THE ELLIPTIC FUNCTION fp(z). 327 This differential equation can be written in the form where ^ »= j? (^)i and therefore (since ^ (z) is infinite when z is zero) we have which is the required expression of ^(z) in terras of an integral. The preceding theorems may be illustrated by the results which correspond to them in the theory of the circular functions. Thus we may in the following way discuss the properties of a function f{z) (really ca$ec^z)f which we shall take to be defined by the series This series is clearly infinite at the points ^^O, ir, - ir, 2ir, ... ; for other values of z it is absolutely and uniformly convergent, as is seen by comparing it with the series H-l-«+l-«+2-a+2-«+3-«+3-*+.... The effect of adding any multiple of ir to ^ is to produce a new series whose terms are the terms of the original series, arranged in a difierent order ; this does not affect the sum of the series, since the convergence is absolute ; and therefore/ (^) is a periodic function of X, with the period w. By drawing parallel lines in the 2-plane at distances w from each other, we therefore divide the plane into strips, such that at points occupying corresponding positions in the different strips, /(f) has the same value. In each strip, f{z) has only one singularity, namely at that one of the points 0, ir, — «-, 29r, -29r, ... which lies within the strip. The function is not infinite at the infinite ends of the strip, because the several terms of the series for f{z) are then small compared with the corresponding terms of the comparison- senefi 1 +l-«+l-«+2-«+2-a+3-«+3-«+... . Now near the point f =0, the function /(«) can be written in the form /«='-*+'-'(l-i)"*+'-'(l+^)'*+(2')-'(l-2^) "*+... =*-*+ir-*(l+l + 2-*+8-«+...)+»r-M(3+3+3. 2-« + 3. 2-«+. ..)+.. . =«-»+2«-« . ^+,r-**». 3 . 2 . ^+... Differentiating and squaring this equation, we have It follows that /"W-6/«(*)+4/(*) 328 1*RANSCENDENTAL PUNCTIONa [CHAP. XIV. is a series containing no negative powers of z; it has therefore no singularity at the point ^^■■0, and therefore (since that is the only possible singularity) no singularity in the strip which contains 2=0, and therefore (on account of the periodic property) no singularity in any strip. It is therefore, by Liouville's theorem (§ 47), a constant : this constant must be equal to the value of the function at the point ^--O, which (on substituting the expan- sions) is found to be zero. We have therefore /"W-6/«W+4/(,)-0. Multiplying by 2/' (t) and integrating, we have where c is a constant. On substituting the expansions, c is found to be zero, and therefore /'»«=4/«(i){/(.)-l} or (£)*-^ (^- ^)' ^^^ ^-/W> which gives 2«- / f^{t- 1)"* dt as the expreesioD of f(s) by means of an integral Example, If y = ^ («), shew that \dz) \ck) where «i, «s, «} are the roots of the equation For we have f(^('^)'^«^(')-9fi?(s)^9zf and so (^y-4(y-«i)(y-«,)(y-6j. Differentiating logarithmically, we have ''^-(y-«i)-^+(y-«j)-^+(y-«s)-^. Dififorentiating again, we have ds^ \dz*) \dz) \aSJ Adding the last equation, multiplied by ^, to the 49quare of the preceding equation, multiplied by fj, we have the required result. It may be noted that the left-hand side of the equation is half the Schwartzian derivative of z with respect to y ; and hence the result shews that z is the quotient of two solutions of the equation 179, 180] THE ELLIPTIC FUNCTION ff (z). 329 179. ITie homogeneity of the function fp (z). When the Weierstrassian elliptic function is considered as depending on its arguments and periods, it has a certain property of homogeneity, which will now be investigated. Let fp Iz, M denote the function formed with the argument z and periods 2a>i and 2^2. Then we have It follows that the effect of multiplying the argument and the periods by the earns quantity \ is equivalent to multiplying the function by X"^. This relation can also be expressed in terms of the quantities gtt g^ For let fp(z; g^, g^) denote the function formed with the invariants g^ and g^. Then we have g^ss 60 2 (2mo)i + 2w<tt,)"*, ^fj =s 1402 (2mft>i + 2nG)a)~*. The effect of replacing a>i and o>s by Xq>i and Xto^ respectively is therefore to replace g^ and g^ by X^g^ and X~*5r, respectively; and thus we have K';s„}.)-f{'. ^) M"- ^ = X»jf>(X(r; X-*g„ \-*gt), m which expresses the homogeneity-property in terms of the invariants. Example. Deduce the last result directly from the equation 180. The addition-theorem for the function fp (z). The function fp(z) possesses an addition-tiieorem, i.e. a formula which gives the value of |> (xr + y) in terms of the values of fp (z) and fp (y), where JB and y are any quantities. 4f •4 830 TRANSCENDENTAL FUNCTIONS. [chap. XIV. To obtain this formula, consider the expression ,1 F(*) 9'^') ! 1 «»(y) F'(y) as a function of z. Since it is compounded of doubly-periodic functions, it is itself a donbly- periodic function ; and the only points at which it can have singularities are the points at which the functions jp(^ + y) and ^{z) have singularities, i.e. the points ^ = 0, z^^y, and points congruent (§ 177) with these. Now for points z near the point ^ = 0, we can write the determinant in the form 1 ip(y) + ^|)'(y) + W'(y) + ... -i?'(y)-^j?"(y)-... ^+Mfl^«^+ — -2r-»+j^flra« + .., 1 viy) ip'(y) Expanding this determinant, we find that the terms involving negative powers of z destroy each other; the determinant can therefore, in the neighbourhood of the point ^ = 0, be expanded as a series of positive powers of z ; that is, the function represented by the determinant has no singularity at the point z^O; and therefore (by the periodic property) it has no singularity at any of the points congruent with z^O. Considering next the neighbourhood of the point z^ — y, write ^ = — y + a?. The determinant can be written in the form 1 jf>(y) «>'(y) and on expansion this is found to contain no negative powers of x. The function represented by the determinant has therefore no singularity at the point xr = — y or any of the congruent points. The function has therefore no singularities, and so by Liouville's theorem (§ 47) is independent of z. But it vanishes when z has the value y, since two rows of the determinant are then identical.. The determinant is therefore always zero. We thus have the formula 1 ^{z + y) -p'(xr + y) 1 «>(y) v'iy) = 0, 180] THE ELLIPTIC FUNCTION ^{z). 331 true for all valnes of z and y. Since, by § 178, jf^ (^ + y)> §>' {^\ j?'(y) ^re at once expressible in terms of jp(-f + y), jp(^), |>(y), respectively, this result really expresses ^{z-k-y) in terms of ^{z) and jf>(y). It is therefore an addition-theorem. The addition-theorem may also be obtained in the following way. Take rectangular axes Oxy Ou, in a plane ; and consider the intersections of the cubic curve with a straight line The abscissae Xi, x,, x, of the points of intersection are the roots of the equation <^(x)=0, where <p (x) « {mx + n)« - 4c5 +^^ +^,. The variation liXr in one of these abscissae, consequent on small changes dm and dn in m and n, is therefore given by the equation <f/ (Xr) 4x^+2 (mxr + n) (Xi^w + dn) ■■ 0, whence » to. ft i x-dm + d» — 2 2 Therefore bO, by a well-known theorem in partial fractiona 8 Now when n is infinite, the abscissae x^, x^, x^ are all infinite : we may therefore int^;rate the last equation over the series of positions of the straight line ytamx-^-ny and obtain the result 3 /•• 2 I {AXr^^-g^r-g^'^dXr^O. r-1 y aL If we write *i-j?W, *a=j?(y)» *8-if>(t«^), we have therefore z •\-y + tr ■■ 0. But the ordinates of the three points of intersection are «,-!>'(,), Wj-ip-Cy), «,-if)'(to). Since the three points are collinear, we have I ^3 W3 -0, I jp(y) r(y) which is the addition-theorem. and therefore 332 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV. 181. Another form of the addition-theorenL The determinantal form of the addition-theorem given in the last article may be replaced in the following way by a simpler, though less symmetrical, formula. Consider the equation 1 jp(a.) f(/(w) =0. 1 j?(y) jf>'(y) If in this we replace fp> {x) by its value in terms of |> (x), and expand, we have {4t>» (w) ^g,fp (x) - g,\ { jp (z) - 1> (y)}« = [fp' W {p (^) - j? (y)} + jf>' (y) {» (^) - 1> (^)}?. This may be regarded as a cubic equation in the quantity |> (x). One of its roots is jp(^) = |>(^+y), by the addition-theorem; and the other two roots are jp (^) = jp (z) and fj>{x)^jp (y), since the determinant vanishes when x^ or y is substituted for x. We have therefore |> (z) + |p(y) + ip(^ + y) = Sum of roots of cubic = - {Coefficient of p«(a;)} -r {Coefficient of f^(x)} = i {J?' (^) - j?' (y)}M«> W - j? (y)}"*, and thus we have which is a new form of the addition-theorem. Example 1. Prove that the expression i{r W - r (y)}' {J? W - f (y)} -* - «» W - f («+y). considered as a function of z^ has no singularities: and deduce the addition-theorem for ip(z). For the given expression, from the mode of its formation, can clearly have no singu- larities except at the points e=xO, z=yy z^-y, and points congruent with these. Consider then first the neighbourhood of the point z—0. The expression can be expanded in the form -P(y)-«r(y)--. and this on reduction is found to contain no negative powers of e, the first non-zero term being fp (y). The expression has therefore no singularity at the point ^=0. 181, 182] THE ELLIPTIC FUNCTION ^{z). 383 Considering next the neighbourhood of the point z^y^ we take fny+x ; the ezpressioQ becomes i {fr' (y)+*r (y)+ - - r (y)}Mf (y)+««>' (y)+ - - P (y)}-'- if (y) -*r (y) - - -I»(2y)-xif>'(2y)-..., and this on reduction is found to contain no negative powers of x ; there is therefore no singularity at the point z^y. The case of the point z^ — y can be similarly treated. ■ The given expression has therefore no singularities, and so by Liouville's theorem is independent of z. But its value at the point z^O has been shewn to be ^ (y). We have therefore, for all values of Zy i{P''w-r(y)}*{i»W-jP(y)}-*-«>«-P(*+y)-«'(y)-o, which is the addition-theorem. Example 2. Shew that P(«+y)+«>(«-y)- {if>«- jp(y)} -»({2PW (PW-igt) {|»W+i»(y)}-s'J. For by the addition-theorem we have Replacing fp^{z) by 4pW-5r,p(z)-^„ and replacing f^{y) by 4jf>8(y)-^,if>(y)-^3, and reducing, we obtain the required result. 182. The roots e^, e,, e^. Let n denote any one of the periods of g> {z), namely the quantities 2a),, 2a)3, 2q>i + 2ft)a> 2a)i — 2g)2> "- 2a)i — 2ca2, .... Then §>' (^ fi^ = j>' Q ft - n) , since jf^ (^) has the period ft, = — p' ^2 ft j , since |>' is an odd function of z. It follows from this that unless 2^ ^^ itself a period (in which case jp'T^ftj is infinite), jp'Uftj is zero. We have therefore |>'(«i) = 0, jf>'(«,) = 0, i?'(a>,) = 0, where &>« stands for — (oh + ««>2)- 334 ' TRANSCENDENTAL FUNCTIONS. [CHAP. XIV. Now denote the quantities jf>(a>i), |>(a>s), jf>(G)«) by ^, e,, «,, respectively. Then the equation jp'» (a)i) = 4|l* (a)i) - 5r, jp (o),) - ^r,, or 0=4^»-5r8ei-5r,, shews that ^i is a root of the cubic equation Similarly e^ and ^s are roots of this equation. Moreover, the quantities ei, e,, e, are distinct roots of the equation; for if for example we had 61 = ^s, we should have fp (q>i) = fp (q>i), and therefore o), = ± 0)1 + a period, which is not the case. We see therefore that the three roots of the cvhic are ei, eg, c,, where ei^ip (ft>i), «i = j? (wi), c, = p (o),), and ft)i + ft>2 + ®j = 0. The quantities ^1, es, ^ therefore satisfy the relations ei + «, + es = 0, _ 1 1 183. Addition of a half period to the argument of ^ (z). From the addition-theorem we have ip{z + G)0 + ip(2) + 61 =5|>'*(^) {jf>(^) - eij-« = {if> (^) - (h] lip (z) - e,} {^ (z) - ex}-S or j?(^ + fth) = ^ + («i - «i)(^ - <?i) (ipW - «i}"'. This formula expresses the result of adding a half-period to the argument of the Weierstrassian elliptic function. 183, 184] THE ELLIPTIC FUNCTION ^ (z), 335 Example 1. Shew that is a multiple of the dlBcriminaut of the ec[uation For we have jf> (Z + »i) - «! = (61 - «jj) («! - «3) {P («) - <H}-1. Differentiating, we have Therefore - 16 (ej - e^y (<52 - tfj)* (63-61)2, which is a multiple of the discriminant of the equation 4 (^-61) (^—62) (^-6^=0. Example 2. Shew that {jP(2^)-«i}{j?(2«)-«J+{«>(2^)-«i}{i?(2*)-6,}+{P(2»)-eJ{p(2^)-63} = |>(«)-^ 184. Integration of (oa?* + 46^;* + Qca^ + 4(fo? + eY^. We shall now shew how certain problems in the Integral Calculus, whose solution cannot be found in terms of the elementary functions, can be solved by aid of the function |> {z). Let the general quartic polynomial be written f{x) = our* + Aba^ + 6ca? + ^dx + e. Let its invariants* be g^sioe — 46d + 3c*, 9t^ a b c bed c d e = ace + 2bcd - c» - (wP - i'^e ; let its Hessian be A(a?)«(ac - 6«)a?* + 2 (ad - 6c)a;» + (ae + 2bd - S<^)a^ -4- 2(6e - cd)a? + (c6 - cP), and let its sextic covariant be t(^)^l {-/(^) A' (^) + A (^)/' (^)} = (a"d - 3a6c + 26«)aJ«H- .... * The student who is not already familiar with the elements of the theory of binary forms is referred to Bomside and Panton's Theory of Equatiom^ where the invariants and oovariants of the quartio are discussed. 336 TSANSCENDENTAL FUlfCTION& [CHAP. XI7. Then it is known that <• (^) = - 4A' {x) + g,f* {x) h (x) - (7,/« (x). If we write « = — A (x)//{x), this relation becomes i^{x)^f*{x)i^-g^-g,). Now ^,M.)/-q-y (.)/(.) ^ and so (4«' — 5^^ — ^r,)"* cfe = 2 {/(a?)}"* cte. Let Xq be any root of the equation /(w) =» ; then to the value a? = o!^ corresponds « » 00 ; and hence, if we write z=r{f(x)}-idx, we have 2z « / (4^ — g^ — fftY^ dt. It follows that ths eqimtion »{^z\ 9z. 9t)^-h{x)lf{x) 18 an integrated form of the equation = / {cur* + 4iba^ + Qcaf^ + ^dx + e}-*(fo?. z Example I. Shew that (with the same notation) P'(2«;ir,.fl',)-T<(*){/(x)}-i. Example 2. Shew also that, if then fp (z-i-y) and |>(^~y) are the roots of the equation where F(Xy u)—(M^* + 26a««(47+w)+c(:F*+4PM+w*)+2</(4;+v)+e, and H {x, u) is derived from h (x) in the same way as F{Xj u) from/(j?). (Cambridge Mathematical Tripos, Part II, 1896.) 186. Another solution of the integration-problem. The integration discussed in the last article may also be effected in the following way. As before, let z=r{/(x)}-idx. J xo 185] THE ELUPTIC FUNCTION |> (^). 337 where f{x) =^00* + Aba? + Qca? + 4dx + 6, and let ^0 l>e a root of the equation /(ar) = 0. Then, by Taylor's theorem, we have f{x) = (^ - x,)f {x,) +\{x- x,yr M + g (^ - a:.yr M + i (^ - ^oYr" M Writing (x — Xq)"^ = f» we have / w = i-* {/ ('t-.) ?• + \ f" i^o) ? + 1 r {?=.) r + i /"" (^ . and 80 * = /*{/' (^.)(? + 5/"(^,)r' + 6/"'(^.)? + ^/""(^"*d?. Writing f = 4 {/'(ar,)}-' ^, we have " = /," {4^ + 5 /" (^o) ^ + ^ /' (^•)/"' (^•) ^ + 24 Te/'* C^') /"" (^ ' ^^- Now take a new variable of integration «, defined by the equation this substitution destroys the term involving the square of the variable of integration in the denominator, and we thus have where 5'. = ^/"'(«'.)-h/'(^'>)/"'(«'«)' ^. = h {/' (^o)/" (^•) /'" (^o) - \ /"• (^o) - 1 /'* (^•)/"" H • It can easily be verified that these latter quantities are the same as the invariants g^ and g^ of the last article. We have therefore and therefore ^ "= if^ l-^) "" 24 ^" ^^^* ?=4{/'(^.))-'jif>W-i^/"(^, and finally x = x,^\f' (x.) jjp {z) - ^ /" (x,)} "' . This last equation is the integral-equivalent of the equation z=r{fix)}-idx. w. A. 22 338 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV. It may be observed that $>' (z) = (4s» - <7^ - g,)i = i/' (x,) [/ix)]i ?», and hence that Example, Shew that the integrated form of the equation 'hdx, where ^o ^ ^^7 constant (not necessarily a root of f{x))y and f{x) is any qnartic function of X is where (f> is the Weierstrassian elliptic function formed with the invariants g^ and g^ of/(^). Shew further that 186. Unifomiisation of curves of genua unity. The theorem of the last article may be stated somewhat diflTerently thus : If two variables y and x are connected by an equation of the form y« = cwc* + 4iba^ + 6ca?* + 4(ir + e, then it is possible to express them in terms of a third variable z by means of the equations y = i /' M <»' (^) {fr> (^) - r4 /" (^ " ' where f{x) = aa^ + ^ba^ + 6ca^ + 4da? + e, Xq is any root of th^ equation f{x) = 0, and the function ^(z) is formed with the invariants g^ and g^ of the quarticf{x) ; moreover ^ the quantity z is defined^ by the equation z^\''[f{x)]-^dx. Now y is a two- valued function of x, since the quantity ± (cwr* + 46ar» + 6ca^ + ifdx + e)* 186] THE ELLIPTIC FUNCTION jf>(^). 339 may take either sign ; and x is a four- valued function of y, since the equation in X aa^ H- 46^5* + ^ca? + ^dx + (e - y«) = has four roots. But on referring to the equations which express x and y in terms of z^ we see that x and y are one-valued functions of z. It is this fact which gives importance to the variable z\ zis called the uniformising variable of the equation y* = oar* + 4iba^ + Qca^ + ^dx + e. The student who is acquainted with the theory of algebraic plane curves will be aware that curves are classified according to their genvs*^ a number which may be geometrically interpreted as the difference between the number of double points possessed by the curve and the maximum number of double points which can be possessed by a ciuve of the same degree as the given curve. Curves whose genus is zero are called unicursal curves ; if f(Xy y)=0 is the equation of a unicursal curve, it is known that x and y can be expressed in the form » where <f> and ^ are rational functions of their argument ; since rational functions are always one-valued, it foUows that the variable t thus introduced is the uniformiting variable for the equation /(j;, .y)=0 ; i.e., although y is in general a many- valued function of x, and x is a many-valued function of y, yet x and y are one-valued functions of z. Considering now curves whose genus is not zero, let f{x,y)^0 be a ciurve of genus unity. Then it can be shewn that x and y cau be expressed iu the form 'x=<^) {z) where and ^ are now elliptic functions of their argument z ; x and y are thus expressed as one- valued functions of 2, and z is the uniformising variable of the equation /(^, y)=0. This result is obtained by writing where ^and O are rational functions of their arguments, and choosing ^and O in such a way that the equation /(:>;, y)KO is transformed into an equation of the form we can then wnte and X and y will thus be expressed as one- valued functions of z. When the genus of the algebraic curve n^yy)=o is greater than unity, the uniformisation can be effected by means of automorphic functions. Two classes of automorphic functions are known by which this uniformisation * In French genre^ in German OetchUcht. 22—2 c 340 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV. may be effected : namely, one which was first given by Weber in Qdttinger NackriclUen, 1886, and one which was first given by the author, Phil. Trans., 1898. In the case of Weber's functions, the " fundamental polygon " (the analogue of the period-parallelogram) is "multiply-connected," i.e. consists of a region containing islands which are to be regarded as not belonging to it. In the case of the functions described in Phil. TVatis., the fundamental-polygon is "simply-connected," i.e. is the area enclosed by a polygon. This latter class of functions may be regarded as the immediate generalisation of elliptic functions. Miscellaneous Examples. 1. Shew that «>(^+y)-j?(«-y)=-rwr(y){i?W-i?(y)}-*. 2. Prove that where, on the right-hand side, the subject of differentiation is symmetrical in «, y, and w. (Cambridge Mathematical Tripos, Part I, 1897.) 3. Shew that r'(^-y) r'(y-^) r'o^-«) r(^-y) r(y-^) r(^-^) P(^-y) j?(y-tz^) i?(t^-«) =i^2 r'(^-y) r'(2/"^) f?"'(w-z) iPi'-y) PCy-^^) ip{w-z) 11 1 (Trinity College Scholarship Examination, 1898.) 4. If simplify the expression where c^, e.^, e^ are the values of jf> (z) for which J>' («)==0. (Cambridge Mathematical Tripos, Part I, 1897.) 5. Prove that 2{PW-«}{iP(y)-i?W}Mj?(y+^)-«}*{j?(y-^)-4*=o, where the sign of summation refers to any three arguments z, y, w, and e is any one of the quantities ^i, e2) ^s* (Cambridge Mathematical Tripos, Part I, 1896.) 6. Shew that £i?i^i)= _|PJK)-#> W. (Cambridge Mathematical Tripos, Part I, 1894.) 7. Prove that if> (2«) - p(a,,) = {j!)' (^)}-« {if> w- f> (K)P (F «- P (»»+K)}*- (Cambridge Mathematical Tripos, Parii 1, 1894.) MISC. EXS.] THE ELLIPTIC FUNCTION fp {z). 341 8. If m be any constant, prove that ^l r r e^{fp{z) - ft (y)} p (^) ^^^y " 2^J; {i?«-^i}{i?(y)-^i} ' where the summation refers to the values of jp («) for which jp' (^) is zero ; and the integrals are indefinita (Cambridge Mathematical Tripos, Part I, 1897.) 9. Let and let (s0 {x) be the function defined by the equation where the lower limit of the integral is arbitrary. Shew that _J^Js^ ^ »^(a+y)+<^^(«) , 4>'{a-y) + 4>'{a) _ <i>'{a'¥y)-<i>'{x) *(A'+y)-*(«) </>(a+y)-*(«) </>(a-y)-*(«) *(«+y)-*(^) <»-(a-y)-<^-(:r) ^ (Hermite.) 10. Shew that when the change of variables is applied to the equations rfw_. i^l — _=0, 2iy4-l+K they transform into the similar equations 2i7'+l+^f Shew that the result of performing this change of variables three times in succession is a return to the original variables |, i; ; and hence prove that if | and i; be denoted as functions of u by E(u) and F{u) respectively, then where A is one-third of a period of the fiuictions E(u) and F{u), Shew that • iJ^(t*)=^- j? (w ; gi> ffz\ 12 where ^s-^p+lgl^, 5^3= -1-^^- 216^' (De Brun.) CHAPTER XV. The Elliptic Functions snz, cnz, dnz, 187. Construction of a doubly-periodic function with two simple poles in each period-parallelogram. The function ^{z\ which has been considered in the previous chapter, is a doubly-periodic function of z, with a single pole of the second order in each period-parallelogram, namely at the point congruent with the origin*. We shall next introduce a doubly-periodic function which differs from ^ (z) in having two poles, ecwh simple, in every period-parallelogram. Consider the series /(z) = S [{^ + 2mfi)i -h (2n + 1) to^}-^ - {2m^, -h (2n -h 1) ao^}-^ - {2r -f- (2m -h 1) ft), -h (2n -h 1) ft)^}-* -I- {(2m -h 1) ft>i + (2n -f- 1) a>»}-^, in which the summation extends over all positive and negative integer and zero values of m and n. When the modulus of (2ma)i -h 2nft)j) is large (and we may suppose the series arranged in order of ascending values of |2ma)i-h2nft),|), the terms of the series bear a ratio of approximate equality to those of the series 2 [- 5 [2mwi H- (2n -h 1) ft),}-« -h z {(2m -h 1) ft), -h (2w -f- 1) ft),!"*], or — ^ 2 {2mft), -|- (2n -h 1) ft>a} i-a..--- -.,-n. Ol 2ma)i +(2n-|- I)ft)al and these terms bear a ratio of approximate equality to those of the series - 22rft)i X {27nft)i -h (2n -h 1) ft)^}-*, which again bear a finite ratio to those of the series S (2mft)i -h 2nft)a)~', which was shewn in § 11 to be an absolutely convergent series'. • In the network of paraUelograma described in § 177, the poles of ^ {z) are not within the parallelograms, but on their bounding lines. We may however suppose the whole network slightly translated so as to bring the poles within the parallelograms. 187, 188] THE ELLIPTIC FUNCTIONS sn Z, CD 2, dn z. 343 It follows that the sertes which represents f{z) is absolutely convergent for all values of Zy except for the exceptional values included in the formula z = ma>i + (2w + 1) (Wj, (m, w, integers) for which the several terms of the series are infinite, and which have been tacitly excluded from the foregoing discussion of convergence. Moreover, since the terms of the comparison-series are independent of z, the convergence is (§ 52) not only absolute but uniform. By a discussion similar to that in § 176, we can shew that/(2r) is a dovhly- periodic function of z, whose periods are 2(Oi and 20)3 ; it is an odd function of Zy so that f{z) = -n-z); and its singularities are at the points z = m<Oi + (2n + 1) wg, where m and n may have any integer or zero values; these singularities are simple poles, with the residues + 1. There are two of these singularities in each period-parallelogram. 188. Expression of tJte function f(z) by means of an integral. The singularities of f{z) in the fundamental period-parallelogram are, as we have seen, at the points z^a)^ and ^ = (Wj + oj. Consider now the neighbourhood of the point z = ©j. Writing -8^ = ft)j -I- ic, we have /(ft)j + a?) = — /(— 0)2 — a?), since / is an odd function, = — /(2ft)3--6)j — a?), since 2ft)2 is a period, = -/(6)2 - a?), from which it follows that /(oij + a?) is an odd function of x ; the expansion oi f{z) in ascending powers of x will therefore contain only odd powers of a?. Now f{z) » S [{a: + 2m(o^ -h (2n -h 2) to^]"^ - [2m(^ -|- (2n + 1) «,}-* - {a? -h (2m + 1) Oh + (2w -h 2) ©3}-^ + {(2m -h 1) ©i + (2n -h 1) <o^'% where the summation extends over all positive and negative integer and zero values of m and w. In this expression, replace all expressions of the form (-4 + x)"^ by their expansions A"^ — A~^x -h A^^a^ — . . . , x being supposed small. A term 944 TRANSCENDENTAL FUNCTIONS. [CHAP. XV. in x"^ will arise fix)m the pair of values (m = 0, n = — 1), and we thus have w where 5 = 2 [- [2mwi + 2wg)j}-« + {(2m -h 1) ©i + 2/wtfa}-«] . the summation being in this case extended over all positive and negative integer and zero values of m and n. excluding simultaneous zeros in the first term. If now by means of this expansion we express the quantity as a series of powers of x, it is found that the negative powers of x destroy each other ; this quantity has therefore no singularity at the point z = Wa. Consider next the neighbourhood of the point 2: = wi + ©j. Writing 2: = ©i + oja + y, we have /(©i -h 6), + y) = — /(— ft)i — o>j — y), since / is an odd function, = —/(cDi-hwa— y), since (2fth + 2ft)2) is a period. It follows that f((Oi + Wa + y) is an odd function of y ; its expansion in powers of y will therefore contain only odd powers of y. Now expanding/(ir) in powers of y, in the same way as /(^) was formerly expanded in powers of x, we find that /(z) l + B'y + Cy + ..., where 5' = S [- {(2m - 1) ©i + 2r?6)2}-» -h {2mG)i + 2nft)j}-«], the summation extending over all positive and negative integer and zero values of m and n, excluding simultaneous zeros in the second term. (Comparing this with the expansion of B, we have so /(^) = -l-5y + ay + ..., and, as before, the quantity has no singularity at the point z — fOx-\- ci>a. Now the points z = 6), and ^r = ©i 4- <»j are the only possible singularities of this quantity in the period-parallelogram ; it has therefore no singularity in the parallelogram, and therefore (since it is doubly-periodic) no singularities 189] THE ELLIPTIC FUNCTIONS 811 ^, cn 2r, dn ^. 345 in the whole -er-plane ; it is therefore by Liouville's theorem (§47) a constant independent o( z, say A. The function /(^) therefore satisfies a diflFerential equation Replacing B and A by new constants k and /t, we can write this in the form /'»(^) = {^,-/'(^)}{;^,-/M^)}; SO that, Baf(z) is zero when z is zero, -ris-") "'a.-rt dt. We see therefore that the odd doubly-periodic function f(z), which has periods 2ft)i and 2(d^ and simple poles at all points congruent with ^ = oh and -8^ = ft)i + 0)2, may he regarded as defined by the equation cnz)(]^ wji w where k and fi are constants depending only on cdi and Wj. 189. The function sn z. The function f{z) discussed in the last two articles can be expressed in terms of another function, which we shall denote by sn z, in the following way. Replacing the variable t of integration by a new variable «, defined by the equation ks = fU, we have zr^fil (1 - «»)-*(! - *»«>)-* d«. Jo Now define the new function sn -e by the relation fifijjbz) = kanz; then we have runM Jo This last equation can be regarded as the definition of the function sn z m terms of its argument z and the constant-parameter k, which is called the modulus ; it is analogous to the definition of the function sin z by the relation rwnz = / (1 -«»)-* ds. 346 TRANSCENDENTAL FUNCTIONS, [CHAP. XV. From the equation fif(jiz) = & sn ^, it is clear that the function sn z has the same general properties as f{z)y namely, it is an odd one-valued doubly-periodic function of z, with two poles in each period-parallelogram, the distance between the poles being half of one of the periods. The two periods will be connected by a relation, as they depend only on the single constant h 190. The functions en z and dn z. We now proceed to introduce two other functions, either of which may be regarded as bearing to the function sn^ a relation similar to that which the function cos z bears to sin z, Csaz Since z^\ (1 - «»)-* (1 - k"^)-^ ds, Jo dz we have -r? c = (1 — sn* z)~^ (1 —k^ sn' z)~^, a{snz) ^ ^ ^ or -T- (sn -^) = (1 - sn' z)^ (1 - A:" sn» z)K Now sn ir is a one-valued function of z, so its derivate must be also a one- valued fimction. It follows that (l-sn'^)*(l-A'sn«2r)* can have no branch-points (§ 46), considered as a function of z ; and therefore either (a) Elach of the quantities (1 — sn'^r)* and (1 — Ar'sn'^r)* is a function of z which has no branch-points, or (/8) The functions (1— sn'z)* and (1— A'sn'^)* have branch-points, .but are such that their product has no branch-points. Now the alternative (fi) could be true only if the functions (1 — sn' z)^ and (1 — Ar'sn'^)* had their branch-points at the same places ; but this is not the case, since (1— sn'^)* has branch -points at the places when sn'2: = l, and (1 — A'sn'^)* has not. The alternative ()8) being thus ruled out, we see that the alternative (a) must hold. If now we write en 2: = (1 — sn'^:)*, dn;r = (l — i'sn'-e)*, where it is supposed that each of these functions has the value unity when sn z is zero, then since en z and dn z have no branch-points, and have definite values at the point 2: = 0, it follows that the functions en z and dn z are one- valued functions of z. 190, 191] THE ELLIPTIC FUNCTIONS 811 ^, cn z, dn z. 347 They obviously satisfy the relations an^z + cn^z= 1, l^sn^z + dn'2:= 1. The functions sn z, en z, dn z are often called the Jacobian elliptic Ainctions, The function cos z is in the same way a one- valued function, although the occurrence of the radical in (1 -sin'^)' might lead us at first sight to suppose that it possessed branch- points. 191. Expression of en z and dn z by means of integrals. We shall next find, for the functions en z and dn z, integral-expressions similar to that found in § 189 for sn^^. Diflferentiating the equation cn*z = 1 — 8n*z, we have en -^ -y- en -? = — sn .8^ en z dn -s^, dz so -7- en -? = — sn -g' dn z dz = - 1(1 - en- z) (*'« -h k" cn« z)]^, where &'* = 1 — A;*. Thus ifcn^ = ^ we have d^ = - (1 - <»)-* (*'« + h^t")-^ dt, and therefore (since en ^ = 1 when z = 0) z^r (l-.t«)-*(A'»-hifc»t')-^(ft. In the same way we can shew that J- dn -? = — A^ sn z en z, dz and z=r (l-^)-*(t«-A/»)-*d«. J dm Example 1. If cs 2»cn ^/sn z^ shew that J C8« Example 2. If sd 2 -> sn zjdn ^ shew that - {"^ \\ - h'H^yk (1 +i:»^)-i efe i 348 TRANSCENDENTAL FUNCTIONa [CHAP. XV. 192. The addition-theorem for the function dn z. We shall next shew how to find dna:, where in terms of the sn, en, and dn, of y and z : the result will be the addition^ theorem for the function dn. Suppose that y and z vary, x remaining constant, so that ^^ — — 1 * Introducing new variables u and v, defined by the equations w = en ^ en y, t; = sn ^ sn y, we have or dv 1 . ^ d^ , -T- sn 2: en V dn V + sn V en 2r dn -er -y- dv _d£^ ^ ^ ^ ^ dy du du \ \ az T- — en z sn y dn y — en y sn -8^ dn ^ -1- dv __ sn 2r en y dn y — sn y en ^ dn ir du Gnysazdnz—cnzsnydny' From this we obtain the equations (-T-) - 1 =A:*(sn'y — 8n*2r)'(cny sn ^dn 2r — cn^rsny dny)"*, dv v^u-T- = (sn y en y dn ^ — sn 2: en ^ dn y) (en y sn j dn 2: - en 2: sn y dn y)""^ (-r-j ■~(^"'^;7~) =(sn'y--sn'^)'(cny sn^dnz — cn2:8n ydny)"*, and consequently L /"^V -. 1 - /"^ V ( - ^^V i^Kdu) k'^Kdu) V ^'dul' or *■(«-"*■)'—*•■©■■ This equation is the equivalent, in the new variables, of the equation dz - dy" It is a differential equation of Clairaut's type, and its integral is therefore ifc» (t; - -Mc)* = 1 - A: V, where c is an arbitrary constant. 192] THE ELLIPTIC FUNCTIONS sn ^, CH Zy dn z, 349 Thus the equation A» (sn 2r sn y — c en -^ en y)* = 1 — AV must be equivalent to the equation where c is some funetion of x. To determine c in terms of x, put y = ; then we have ifc«c* en* a; = 1 - Jk V, which gives c* = dn"'^; = dn~* (z + y). Now the integral equation can be written in the form c» (1 — Jt» -f jfcj en' y en' ^) — ick^ sn y sn 2: en y en ^r + (A:* sn' y sn' 2r — 1 ) = 0. Solving this equation in c, we have _ A^'sny sn 2: en y en ^ t {Ar* sn'y sn'^r en'y en* ^—(1— Ar'-I-A:' cn'yen*2:)(A:'sn'y sn'^—1 )}* " 1 — A* -h A* en* y en* ^ _A:*snysn2renycn^±dnydn^ ^^ ^" 1-A* + A;*en*yen*^ ' Sinee A:* sn* y sn* 2: en* y en' ^ — dn*y dn* = (1 — A;* + A* en* y en* 2:) (A:* sn* y sn* ^ - 1), this equation ean be written A:* sn* V sn* ^ — 1 or 4* sn y sn 2r en y en 2r T dn y dn ^ * , . . ±dnydn^iA:*snysn2:enyen^ an \Z + y) = = — ^i — i z . ^ ^' 1 — A:»sn*ysn*z The two ambiguities of sign in this equation remain to be decided. Taking ^r = 0, it is seen that the first ambiguous sign must be + ; so J . . dn ?/ dn ^ + A^ sn V sn 2: en V en ^ dn(2r + y)i= — ^ -:=— — f — ^ . ^ ^ 1 — &*sn*ysn*-2r Now suppose that y is a small quantity ; expanding both sides in ascend- ing powers of y, and retaining only the terms involving the first power of y, we have dnz-^-y-^dnz^dnz ±I^8nzcnz, Sinee -j- dn ^ = — i* sn 2? en jg, dz 360 TRANSCENDENTAL FUNCTIONS. [CHAP. XV. it is clear that the ambiguous sign must be — . We thus finally obtain the addition'theorem for the function dn, namely , , . dn-edny— Ar^sn^sny cn^cny Example 1. Shew that ^' ^ ^' 1 - ifc* an* y sn* « Example 2. Prove that , J „ 2dn2^ l+dn2z 193. The addition-ikeorems for the functions sn z and en z. To obtain the addition-theorem for the function sn ^, we have sn(ir + y)=±j^{l-dn«(«+y))» Substituting for dn {z + y) from the result of the last article, this equation after some algebraical reduction gives sn . . _ sn ^ en y dn y + sn y en 2: dn -gf (^ + y) - ± - l^^Jc^^^^Jsn^ • On putting y = in this formula, it is seen that the ambiguous sign is -h ; we thus obtain the additton-theorem for the function sn, namely I V _ sn -8^ en y dn y + sn y en 2r dn -? sn(^ + y)- —-,^^——— . Similarly for the function en z we obtain the addition-theorem . . _ en ^ en y — sn 2r dn ir sn y dn y ^""^^■^^^ 1-A;«sn»^sn^y * These results may be regarded as analogous to the addition-theorems for the circular-functions, namely sin (-3^ -h y) = sin z cos y + cos z sin y, cos (-8^ + y) = cos z cos y — sin ^ sin y, to which, indeed, they reduce when k is put equal to zero. Example 1. Prove that sn(z4-y)sn(«-y)=:; — rs — 5 — o ^ > Example 2. Shew that , 1 - en 2jj 1 H-dn 2« 193-195] THE ELLIPTIC FUNCTIONS SD Z, CU Z, du Z, 351 194. The constant K, We shall denote the integral r(i-eo-*(i-jfc»e«)-*ctt by if; it is clearly a constant depending only on the modulus k. The ambiguity of sign in the radical will be removed by the supposition that at the lower limit of integration the integrand has the value 1. From the equation z 'o we see that sn ir= 1, and hence en if = (1 — sn^ K)^ = 0, Example, Prove that dnir=(l-ifc»8n«£')* = A?'. 196. The periodicity of the elliptic functions with respect to K, It will now appear that the constant K is intimately connected with the periodicity of the elliptic functions sn z, en z, dn z. For by the addition-theorem, we have sn -g: en X* dn X* 4- sn ^ en z dn 2r en z sn{z + K):= Similarly cn(z + K)=='-k' ~ l-k'sn'^zsn^K dnz' , sn^: and cln(^-|-ir) = ^ z Hence sn {z + "IK) = j^ / ^ jl = - sn £, and similarly en {z + IK) = — en -3^, dn(2: + 2iO = dn2r; and finally sn (z + 4ir) = — sn (^ + 2ir) = sn z, en {z + ^K) = en z, dn {z + 4i0 = dn z. 352 TRANSCENDENTAL FUNCTIONS. [CHAP. XV. This 4iK is a period for the functions sn z and en z, arid 2K is a period for the function dnz. Example. If cs 2 = en z/an z, shew that cszca(K-z)^t, 196. The constant K', We shall denote the integral The ambiguity of sign in the radical will be removed by supposing that at the lower limit of the integration the integrand has the value 1. Write « = (1 - ib"^)-*. Then (i-^)-*=^(l-A:V)i, and (1-^^)"* = ^^-^^*^ and d« = (1 - k'H^)'^MHdt. 1 Therefore £" = - i\ (1 - «»)-* (1 - h^^Y^ ds, 1 and so ' K-\- iK' = [*(! ~ «>)-* (1 - A^^)-* ds, Jo or m(K + iIC)^Ty whence dn(ir + iZ')=0 and en (ir + tir)= ± ^. To determine the ambiguous sign in the last equation, we observe that the sign of i must be understood in the light of the relation (l-s')-*=^(l-A;'»<')». which was used in the transformation ; putting « = sn(Z + iir') = p «=1. ihf And so en {K + iK) = - -^ . Example. Shew that cni(ir+iJr')=(l-t)(^) . 196-198] THE ELLIPTIC FUNCTIONS sn 2r, en z, dm, 363 197. The periodicity of the elliptic functions vrith respect to K + iK', The quantity K introduced in the last article is of importance in connexion with the second period of the functions snz, cn-gf, dnir. For by the addition-theorem, we have / . IT . -ET/x sn<^cn (^+tX')dn(^ + tir') + sn(-K' + tZOcn<^dn^ Sn (j^ + iL + iJtL ) = ?— ;; j-i TTtt ^TrTl _ dn-er A; en 2:' ik' 1 Similarly en (^ + ^ + iK') = - -r- , xkj sn £ and , dn (^ + iT + iK') = . ^ GHZ By repeated application of these formulae we have sn (z + 2^^ + 2tZ') = - sn-g:, en (^ + 2ir + 2iK) = en z, dn(-^ + 2ir + 2iir) = -dnr, aod fsn(^ + 4ir+4tZ0=sn^, cn(^-|- 4^+ UK') = Gazy dn(^-|-4J5r+4tZ') = dn2r. Hence it appears that the function en z admits the period 2K + 2iK'y and the functions sn z and dn z admit the period 4}K + 4iir'. 198. The periodicity of the elliptic functions with respect to iK. By the addition-theorem, we have sn (^-1- iZ') = 8n (z+ K+ iK'-K) ^ sn (z ■¥ K + iK')cn K dnK - mK en {z '\- K + iK)dn(z + K '\'iK') l-k'sn^K sn^ (z + K-^-iK) 1 ^ ksnz* Similarly we find the equations ^ k snz dn (z +iK) = — i . snz w. A. 23 354 TRANSCENDENTAL FUNCTIONa [CHAP. XV. By repeated application of these formulae we obtain sn (z + 2iK') = an z, en {z + 2iK') = — en -gf, dn{z + 2iK')^--dnz, and (Bn(z + ^K')^snz, en (z + 4dK') = en z, dn (z + 4dK') =^ dn z, 80 that the function sn z admits 2iK' as a period, and the functions en z and dn z admit 4dK' as a period. 199. The behaviour of the functions sn z, cnz,dnz, at the point z = iK\ For points in the neighbourhood of the point 2f = 0, the function saz can be expanded by Taylor's theorem in the form snj? = 8n0 + 58n'0 + 2^sn"0 + ..., where accents denote derivativea Since sn = 0, sn'0 = cnOdnO = l, 8n''0 = 0, the expansion becomes sn'"0=.-(l + A;»), etc. 8nz ^ z "1(1 -{- 1(^) z^ + ... . Hence cnj? = (l — sn'z)* = 1—2^ +..., and dn^ = (l-*»sn«^)* and therefore sn(^ + tX) = -^i-- =5^{l-l(l+A.)^+...f-^ 1 . i + h' 199-201] THE ELLIPTIC FUNCTIONS 811^, cn z, dn z. 365 — i 2k^ — 1 and similarly en (z + iK") = nr- H gjr- i? + ... i 2 — Jfc* and dn(-2r + i-K'') = --+ — f— i« + .... It follows that at the point z = iK\ the functions sn z, cnZfdnz have simple poles, with the residues 1 _i k' k' *' respectively. 200. General description of the functions sn ^, en z, dn ^. Summarizing the foregoing investigations, we can describe the functions sn z, en Zy and dn z, in the following terms. (1) snz is a one- valued doubly-periodic function of z, its periods being 4iK and 2iK\ Its singularities are ^t all points congruent with z^iK' and z = 2-ff' -f iK* ; they are simple poles, with the residues Ar* and — k"^ respectively ; and the function is zero at all points congruent with ^ = and z = 2K, It may be obeerved that no other function than sn z exists which fulfils this description. For if </>(;;) be such a function, then ^ {£) - sn « has no singularities, and so by Liouville's theorem is a constant independent of z ; but it is zero when 2=0, and therefore the constant is zero ; that is. When ](^ is real and positive and less than unity, it is e€wily seen that K and K' are real, and sn z is real for real values of z and purely imaginary for purely imaginary values of z, (2) en ^ is a one- valued doubly-periodic function of z^ its periods being 4iK and 2K-\- 2iK\ Its singularities are at all points congruent with z^iK' and z=i2K -{-iK* \ they are simple poles, with the residues ik^^ and —ik~^ respectively ; and the function is zero at all points congruent with z==K and z^ZK. (3) dn ^ is a one-valued doubly-periodic function of z, its periods being 2K and 4dK\ Its singularities are at all points congruent with z = iK' and z = ZiK' ; they are simple poles, with the residues — i and + 1 respectively ; and the function is zero at all points congruent with z = K + iK' and z^K^2iK\ 201. A geometrical illustration of the functions snz, cnz, dnz. The Jacobian elliptic functions may be geometrically represented in the following way. Let the position of a point, on the surface of a sphere of radius unity, be defined by (1) its perpendicular distance p from a fixed diameter of the 23—2 356 TRANSCENDENTAL FUNCTIONS. [CHAP. XV. sphere, which we shall call the polar axiSy and (2) the angle y^ which the plane through the point and the polar axis (the meridian plane) makes with a fixed plane through the polar axis. Then if ds denotes the arc of any curve traced on the sphere, we clearly have the relation (day = p« {dylry + (1 - p»)-' (dp^. Let a curve (SeiflFert's spherical spiral) be drawn on the sphere, its defining-equation being dy^ = kds, where A? is a constant. We have therefore for this curve and so if « be measured from the pole, or point where the polar axis meets the sphere, we have « = f ' (1 - /)•)-* (1 - A;»p»)-* dp, Jo or p = sn «, the function sn being formed with the modulus k. The rectangular coordinates of the point s of the curve, referred to the polar axis and an axis perpendicular to it in the meridian-plane, are p and (1 — p')*, and can therefore be written sns and cns\ while dn* is easily seen to be the cosine of the angle at which the curve cuts the meridian. Hence if K be the length of the curve from the pole to the equator, it is obvious that sn s and en s have the period 4jK', and dn s has the period 2K. 202. Connexion of the function sn z with the fwnction p (z). We shall now shew how the functions considered in this chapter are related to the elliptic function of Weierstrass. Let ei, ej, e^ denote the quantities ei, ^, ^, taken in any order. Li the integral z^\ I (a? - e^~^{x - ^)-*(a? - e^-^dx, let the variable of integi'ation be changed by the substitution ei-ej a: = e^ + Thus z= {l-t')'^{(ei-ej)-b(ej--et)t'}'^dt, Jo or {ei - 6^)* -8^ = (1 - «•)H^ (1 - l(^t^)-^ dt, Jo 202, 203] THE ELLIPTIC FUNCTIONS sn z, cn z, dn z, 357 where A* = e*-e,- This is clearly equivalent to the equation We thus obtain the result that the function ^{z)yf(yrmed with cmy periods y can he expressed in terms of the function snz by the equation the function sn being formed with the modulus \ei - CjJ Example, Shew that this relation can be written in either of the forms and p(,)=^*-^^5!i(fiZ^}. l-dnM(^-«,)*4 203. Expansion of snz as a trigonometric series. Since sn^r is an odd function of ^, admitting the period 4ir (which we shall for our present purpose suppose to be real), it can by Fourier's theorem be expanded in a series of the foim , . irz , 27r£^ 7 . oirz sn^r = 6, sm 2^ + 6, sm 2^ + 6, sin 2^ + ... , 1 f*^ where (§82) ^'' " ^ I s^^ ^ sin 2K This expansion will (§ 78) be valid for all points in the ^-plane contained in a belt parallel to the real axis and bounded by the lines whose equation is Imaginary part of 2r = ± iK\ since within this belt the function sn z has no singularities. We have now to evaluate the integrals 6^. We shall follow a proof due substantially to Schlomilch. Let OARSCBQPO be a figure in the plane of a variable t, consisting of the rectangle whose vertices are the points 0(t = 0), A(t^ 2K)y C(t^2K+ 2iK')y B{t = 2iK% with a very small semi-circular indentation PQ around the point t = iK\ and another small semi-circular indentation R8 round the point t = 2K -f iK\ 358 TRANSCENDENTAL FUNCTIONS. [CHAP. XV. Consider the integral I sn < « *^ (ft, taken round this contour. Since the integrand is regular everywhere in the interior of the contour, we have (§ 36) \ +1 +1 +( +( +f +f +f =0. J OA Jab J es j so J cb J bq J qp J po Consider first the integral along the semi-circular indentation QP, Writing t = i-fiT' + Re^, we have Jqp Jw irwt /.-« nrJT inr qp 2 w irw snz 2 IT i ^T^ r"2 = T « *^ / (1 + positive powers of R) dO ir% - a rvK' = — r ^ ^^ J when R tends to zero. k Similarly we have imt • rwK' snte^dt^i-iy^e' ^ . Jbs k Since sn (z + 2%K') = sn z, we have J CB J OA* and since sn (z + 2iQ = — sn ir, we have f =(-iyf , and f =(-l)'f . J^tB J PO J 80 J BQ We thus have Now equate to zero the imaginary parts of this equation. Since iiyt sate^^ dt 203] THE ELLIPTIC FUNCTIONS anz, CD 2, dn^. 359 is real when t is purely imaginary, there is no imaginary part arising from JBQ JPO Therefore nrJST' nrJT (l^e^)j^mtBm^dt=le'^{l^(^iy}. 2K k Writing this equation gives IT {\-<f)KK='j^<f{\-{-\r] or and 6y = if r is even. Thus finally we have the expansion of sn ^ as a trigonometric series, sn^ _ 27r / g* irz g* Example, Prove that . ^irz (fi . ^irz A • • • I • Stt ( q^ nz , q^ Zirz . o* bnz ] ^^=2?tri^"^2Z + ii73«>«2^ + l-^co8^ + ..,|. Miscellaneous Examples. 1. Shew that 2. Shew that 3. Prove that 4. Prove that 6. Prove that 6. Prove that ens rdn* dt z /"cn z {l±cn(,+y)}{l±cn(.-y)}=J^^|Bg_ dn*«= ir^-fdn 2g -hitr»cn2g Hrdn2« cn? 360 TRANSCENDENTAL FUNCTIONS. [CHAP. XV. 7. Shew that ^ , dnz-cnz t^+dnz-k^cm" 8. Shew that 8n(*+iJr)=(l+i') S-(1-^)SD«, ' / sn 9. Prove that 8111 [srn-i {sn («+y)}+8m-i {an («-y)}]=^j__-j_-^. 10. Shew that r • 1 f / . M w / xn cn'y— sn*ydn*« co8[sm-i {aD(r+y)}~8m-M8n(^~y)}]= i_^aD»IaD«y * 11. Shew that the quarter-periods K and iK' are solutions of the equation where «=i!*. 12. Shew that the quarter-periods K and tX" are Legendre functions of the argument (l-2ife»), of order -J. 13. Shew that en jScny sn O— y)dn a+cn y en a sn (y- a) dn /9+cn a en j8 sn (a- /3) dn y -l-sn — y) sn (y - o) sn (a— /3) dn o dujS dn y =0. (Cambridge Mathematical Tripos, Part I, 1894.) 14. l{u+v+r+s=Oy shew that i;^ sn t« sn V en r en « - /r> on t£ en t7 sn r sn « - dn t« dn v+dn r dn «=0, iP^ sn t« sn i;- iP^ sn r sn «-|-dn i« dn V en r en « - en tt en vdn r dn tfasOy sn t^sn V dnrdn «~dn ii dn vsnrsn «+cn r en «-ont«cni7sO. (H. J. S. Smith.) 16. Shew that, if a>x>fi>yy the substitutions x~y=(a— y)dn^t« and ar-y=(3-y)dn"*», where it* ■« (a — jS) (o - y) ~ *, reduce the integrals j {{a-'X){x- p){X'' y)}-^ dx and | {(a-x)(a?-/9)(4?-y)}-*d:a? respectively to the forms 2tt (o - y) " ^ and 2v (a - y) " * ; and deduce that, if w + v = JT, 1 - sn^ u - sn* v+ifc* sn* u sn* v=0. From the substitution y= (a- .ir) (a?-/3) («-y)~^ applied to the integral above with the limits p and a, obtain the result W IT 1%! cos* ^ + 6i sin* d) - * c?d = /V* COS* ^ + ft* sin* d) - * <W, where o^, 6] are the arithmetic and geometric means between a and b. (Cambridge Mathematical Tripos, Part I, 1895.) w MISC. EXS.] THE ELLIPTIC FUNCTIONS Sn2r, CU Z, du Z. 361 16. Shew how to exprees I as an elliptic integral of the first kind, in the case when both quadratic expressions have imaginary linear factors. If z^r{(x+l)(x^+x+l)]-idx, express x in terms of z by means of Jacobi's elliptic functions. (Cambridge Mathematical Tripos, Part 1, 1899.) 17. The difierent values of z satisfying the equation cn3z=a are z^, ^, ... z^. Shew that 9 3iH n en Zr+f* U cn«,.=0. (Cambridge Mathematical Tripos, Part I, 1899.) 18. Shew that en « 2ir ( q^ 1FZ q^ ZirZ . q^ birz 1 J = r^ {r" — C0S2rp.-T-^---iC0S^r^+^^— iCOB-r-p— ...}-. auz JcK [l-q 2K 1-^ 2K 1-^ 2K j 19. Prove that k'anz 2ir dnz 20. Shew that 27r f V • «■« Q^ ' 37r2 . q^ . 6rrz ] = 15- (i +y «« 2Z - d^ """ 2^ + 1 +? ""• iz - •••; • J ir 2n ( q itz ^ (fi 2ttz . \ 21. Prove that sn' /l+ifea/TrX 1« /»r\8\ V • ^« fl-Hr>/tr\ 3^ /7r\»l V . "*"t 2;fc3 \^2iS:; 2/fc3 \^2iry / 1-^ ^ 2K Zirz sm — => 2K biFZ sm "^t 2it3 V2i:; 2)E3 V2ir; j i -^^" 2ir (Cambridge Mathematical Tripos, Part II, 1896.) 22. Shew that *« sn2 j= p (a - xK') + Constant, where the Weierstrassian elliptic function is formed with the periods 2K and 2xK'. 23. Shew that the differential equation g={Ji*8n«*-i(l +*»)}« admits the general integral i* = {8ni(C-«)cni(C-a)dni(C-«)}-*{i4 + i9sn«i(C-«)}, where A Mid B are arbitrary constants, and C^2K+%K\ CHAPTER XVI. Elliptic Functions; General Theorems. 204. Relation between the residues of an elliptic function. In this chapter we shall be chiefly concerned with properties of more general elliptic functions than the special functions p (z), sn z, en z, and dn^, which have been discussed in the two preceding chapters. We shall first shew that the sum of the residues of any elliptic function, with respect to those of its poles which are situated in any period-parallelogram, is zero. For let f(z) be an elliptic function, and let icji and 2(0^ be its periods. The sum of the residues is, by § 56, equal to the integral taken round the perimeter of the parallelogram. Now in this integral, any two elements f(z)dz corresponding to congruent line-elements dz on opposite sides of the parallelogram, are equal in magnitude but opposite in sign, and therefore destroy each other. Hence the integral is zero, which establishes the theorem. The number of poles or zeros of an elliptic function contained within a single period-parallelogram is often referred to as the number of irreducible poles or zeros. 205. The order of an elliptic function. We shall next shew that if c is any constant and f(z) is an elliptic function, the number of roots of the equation contained within a period-parallelogrami depends only on f(z), and is inde- pendent of c, and is therefore equal to the number of irreducible zeros, amd also to the number of irreducible poles. 204-206] ELLIPTIC FUNCTIONS i GENEBAL THEOREMS. 363 For the difference between the number of zeros of the function m - and the number of its poles, contained within the parallelogram, is (§ 60) equal to the value of the integral !«•]■ /'W dz 2inj f{z)''C taken round the perimeter of the parallelogram. But if P and Q are two points congruent with each other, situated on opposite sides of the parallelo- gram, then the elements /' {z) [f{z) — c]"^ dz arising from P and Q are equal in magnitude but opposite in sign, and so destroy each other. The integral is therefore zero; that is, the number of zeros of the function f(z)'-o contained within the parallelogram is equal to the number of its poles, i.e. to the number of the poles of 'f{z) ; but this latter number is independent of c, which establishes the theorem. The number of irreducible poles or zeros of an elliptic function is called the order of the function. It must be noted that a zero or pole, which is multiple of order n in the sense of " order " defined in §§ 39, 44, must be counted as n zeros or poles for the purposes of this definition of " order." The order is never less them two; for if an elliptic function had only a single irreducible simple pole, the sum of its residues within any period- parallelogram would not be zero, contrary to the theorem of the last article. This explains why the functions discussed in the two preceding chapters, which are of order two, are the simplest elliptic functions. 206. Expression of cmy elliptic function in terms of ^ {z) and jp' {z). We shall now shew how any elliptic function can be expressed in terms of the Weierstrassian elliptic function which has the same periods. Let f{z) be any elliptic function, and let ^{z) be the Weierstrassian elliptic function with the same periods 26t>i and 2q)s ; and let f^ {z) be the derivate of ^{z). First, we can write f{z)-\ [fiz) + /(- z)] + Y^l^I^ ^' (^). Now the functions f{z)+f{—z) and {/(^) — /(— -sr)} p'~*(^) are even elliptic functions of z\ let {z) denote either of them : we shall now express ^ {z) in terms of (p {z). Since <^(^) is an even function, it follows that if a be one of its zeros in the fundamental period-parallelogram, then another of its zeros in the parallelogram will be congruent to — a : its iiTeducible zeros may therefore be arranged in two sets, say (h» (h>'"CLny and zeros congruent to 364 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI. Similarly its poles can be arranged in two sets, say 61, 62, ... 6n» and poles congruent to — 61, — 6s, ... — 6n« Now form the quantity ~4> (^) { J> (^) - P (6i)} {f> (^) - jf> (6,)} . . . {JP W - i? (6n)} ' where if one of the quantities Or or 6^ is zero, the corresponding factor {pC-^) — j?(M} or { JP (^) -" j? (M) is to be omitted. This quantity is a doubly-periodic function of ^r ; it clearly has no zeros or singularities in the interior of the parallelogram, except possibly at ^^ = 0, and therefore either it or its reciprocal has no singularities in the interior of the parallelogram, and so has no singularities in the entire plane. It must therefore by Liouville's theorem be a constant independent of z. Thus A (z^ - Constant x {<^^^>^ P fa)l W (^) - <f> («.)! » » » {P (^) - P K)) <^(.)- Constant x {p(,)_p(i^)j lj>(.)-|>(6,)} ... {f> (.) - jXtn)}" The quantities {/(^) +/(--?)} and {/(^) -/(--?)} {f>'(^)l-* can thus be expressed as rational functions of ^{z)\ and thus we obtain the theorem that any elliptic function can he expressed in terms of the Weierstrassian function formed with the same periods, the expression being linear in ff {z) and rational in jf>(^). Example. Shew that snzcTkZ^uz^ ^k"^ ^{z- iK'\ where the function p' {z - iK') is formed with the periods 2K and 2iK'<, 207. Relation between any two elliptic functions which admit the same periods. We shall now shew that an algebraic relation exists between any two elliptic functions whose periods are the same. For let f(z) and <l>(z) be the functions. Then by the last article, f{z) and (f> (z) can be expressed rationally in terms of jp (z) and jp' (z). Eliminating jp(z) and p'(z) from the three equations constituted by p''(z) =- i^ip^iz) ^ g,p(z) - g, and these two relations, we have an algebraic relation between /(^) and <(> (z) ; which establishes the theorem. It is easy to find the degree of this equation in / and <}>. For if / be an elliptic function of order m, and if ^ be of order n, then each value of/ determines m irreducible values of z, and each of these determines one value 207, 208] ELLIPTIC FUNCTIONS ; GENERAL THEOREMS. 365 of <^ : so to each value of / correspond m values of <^, and similarly to each value of <t> correspond n values of /. The equation is therefore of degree m in ^ and n in/ Thus IP {z) is of order 2, and jp' (z) of order 3. The relation between them, namely rW=4pW-(73if>(^)-5r3, should therefore be of degree 2 in fp'iz) and 3 in (p {z) — as in fact it is. An obvious consequence of this proposition is that every elliptic function is connected ,tuith its derivate by an algebraic equation. Example. If ty u, v are three elliptic functions of the second order, with the same periods and argument, shew that there exist in general between them two distinct relations which are linear with respect to each of them, namely Atuv-\-Buv+Cvt+Dtu+Et+Fu-^Ov+H=0y A'tuv+B'uv+C'vt+iytu-\-E't+F'u+O'v+E'==0y where A, B, ,,, , H* are constants. 208. Relation between the zeros and poles of an elliptic function. We shall now shew that the sum of the affixes of the irreducible zeros of am, elliptic function is equal to the sum of the affiles of its irreducible poleSy or differs from this sum only by a period. For it f(z) be the function, and 2a)i and 2a)8 its periods, the diflFerence in question is (§ 59) equal to the integral 1 [ zf{z)dz 27rij f{z) taken round the perimeter of the fundamental period-parallelogram. This can be written or or JL r f- {zfS') _ (20,. + ^)/' (26), + ^) ] , 27riLJo t/{^)" /(2a„ + ^) J*" + p L '/'(^ + (2a>, H- y (2., -M) ^1 or ^{-^^'m^^'^'M or 2«^~^**^'®S^"'"^*'''"Sl}. 366 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI. and as log 1 is zero or some multiple of 27rt, the last expression must be either zero or some quantity of the form A multiple of 2a>i + A multiple of 2a)a, ie. a period This establishes the theorem. Example, If F{£) is an elliptic function, for which ^, ^, ... are the irreducible poles, and Ai, A^, ... the corresponding residues, and it f(z) is a one-valued function without singularities in the parallelogram, shew that the integral taken round the period-parallelogram, is equal to 2A^f(z^). (Cambridge Mathematical Tripos, Part II, 1899.) 209. Thefimction f (z). We shall next introduce a function f (5), defined by the equation with the condition that ^{z) — z^^isto be zero when 2r = 0. Since the infinite series which represents |> (z) is uniformly convergent, it can be integrated term by term ; we thus have f (^) = - /" [ir> + S {(-r - 2mfih - 2no>,)-* - {2nm, + Sno^)-*}] dz = -2^* + S {(-r - 2ma>i - 2na>^)-^ + (2ma}i -{■ 2na),)-^ -f z (2mcih + 2nw,)"^}, since the condition, which ^(z) has to satisfy at2r=:0, is satisfied by this choice of the constant of integration. The summation is, as usual, extended over all positive and negative integer and zero values of m and w, except simultaneous zero values. When I 2ma>i 4- 27iq>, | is large (and we can suppose the series arranged in ascending order of magnitude of | 2mo)i + 2na)s i)> the quantity (z — 2mcDi — 2na)8)'"^ -f (2ma>i -f inaj^"^ -f z (2m<0i -f 2na)3)~* bears a ratio of approximate equality to the quantity - z^ (2?nft)i -i- 2yw»a)-». The series which represents ^(z) can therefore be compared with the series S (2nui>i -f 2n<»,)~*, and hence we see that it is absolutely convergent, except at the singularities z = 2ma)i 4- 2na>2, ^^^ that the convergence is uniform. It is evident from the series that at its singularities z = 2mfi)i -f 2wg)2, the function ^(z) has simple poles with residues unity; and that ^(z) is an odd function of z. 209-211] ELLIPTIC functions; general theorems. 367 The function C (') niay be compared with the function cot ;;, whose expansion is cot«=«->+ 2 {(«-wwr)"* + (i?Mr)->}. The equation t- cot z = - cosec* z corresponds to the equation 210. The quasi-periodicUy of the function f(«). Since !>(•* + 2o),) = |> {z), we have _ f (^ + 2a),) = ^ ? (4 or f(^ + 2w,) = ?(«) + 2i7„ and similarly K{^-^ ^Wa) = ?(-?) + 2i7„ where i/i and 172 are two constants introduced by integration. Writing z^ — toi and 2r= — ©j in these relations respectively, we have f (oh) = f (- «.) + 217, = - f (a>0 + 2i7„ ir («.) = f (- ««) + 2172 = - ?(a>2) + 2i;„ whence Vi^K (®i)> which determines the constants rji and 17,. If a:+y+«=0, shew that {fW+f(y)+fW}*+r(^)+rcy)+rw=o. (Schottky.) This result may be regarded as the addition-theorem for the function ( {z), 211. Expression of an elliptic function, when the principal part of its expansion at each of its singularities is given. Lei f(z) be any elliptic function, with periods 26)i and 2a>2. Let its irreducible singularities be at the points z=^a^, ctj, ... On; and let the principal part of its expansion near the point ak be z-au {z-auf '" {z-ajtYh' Then if we consider the function £?(^)=JJct,f(^-a*)-c^f'(^-a*) + ...+j^:^^^ d* where f <*^ (z) denotes j- f (5), we see that 368 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI. (1) When z is replaced by {z + 2a>i), the function E{z) is increased by n n But S Ck is zero, since the sum of the residues o( f(z) within a period- Jb=l parallelogram is zero. Hence E{z) admits the period 2(0i. Similarly E(z) admits the period 2ft),. E (z) is therefore an elliptic function, with the same periods ea/(z), (2) Since the function ^^^^(z — ai) has singularities only at a* and congruent points, and its principal part at a* is (— 1)*** ml (z — ajb)"**""*, we see that E{z) has the same singularities as /(z), and the same principal parts at them. It follows from (1) and (2) that f{z)'-E{z) is a function with no singularities in the whole plane; and therefore, by Liouville's theorem, f(z) — E(z) is a constant. Thus the function f(z) can be expanded in the farm /(2r)=: Constant + 2 S J-^^' c*.f <-« (z - a,). This theorem may be regarded as analogous to the decomposition of a rational function into partial fractions, or the decomposition of a circular function into a series of co- tangents (§ 76). Example 1. Shew that ir sn «= f («- liT') - f («- Sir- tiT') + Constant, where the (-functions are formed with the periods 4 JT and 2iK\ 1 pw rw 1 pw r« Example 2. Shew that j 1 »(y) lf^(y) Extend this theorem to the case in which there are any number of variables. (Cambridge Mathematical Tripos, Part II, 1894.) 212. The function a- (z). We shall next introduce a function ^(z)^ defined by the equation ^log<r(^)=f(2r), with the condition that a (z)lz is to be unity when z=^0. 212, 213] ELLIPTIC FUNCTIONS ; GENEBAL THEOREMS. 369 Since the convergence of the infinite series which represents ^(z) is uniform, the series can be integrated term by term : we thus have log a iz) = log . + 2 {log (l - ^^^ ^ g^ j + 2?rwt)i + 2wa)a 2 (2ma)i + 27k»8)«j ' on choosing the constant of integration so as to satisfy the condition at ^ = ; and therefore the product being, as usual, extended over all integer and zero values of m and n, except simultaneous zeros. The absolute convergence of this product follows from that of the series 1 ^^ V 2ma>i + 2na>2 / "*" 2m^iT2n6^ "*" 2 (2m«i+ 2ni»j)*j ' which is established by comparison with the series - 2 ^ 3 (2ma>^ + 2n(o^y ' since the terms of the two series have ultimately a ratio of equality. It is evident from the product-expression that a (z) is an odd function of z, that its zeros are at the points z^2m(Di + 2na>s> and that z~^a'(z) tends to the limit unity as z tends to zero. The function <r (z) may be compared with the function sin 2, defined by the expansion d The relation ^ log (sin «) = cot « corresponds to ^ log <''(«)" f (^)* 213. The qaasi'periodicity of the function <r {z). On integrating the equation f(^+2ah) = ?(^) + 2i;i we have log a{z-\- 2<o^ = log tr {z)-^ 2%^ + Constant, or ciz-^ 2o),) = C6*»»*^<r (z), where c is a constant. To determine c, write ^ = — oh ; thus <r (w,) = — c^"*'****^- (a)i), or c = — e*»»*^, W. A. 24 370 TRANSCENDENTAL FUNCTIONa [CHAP. XVL and therefore <r (e + 2a)i) = - 6^i («+«.) a {£). Similarly <r (^ + 2(0,) = - e*»«(*+«Oo- {z). The behaviour of the function <r {z) when its argument is increased by a period of (fi{z) is thus determined. By repeated application of these formulae we can find the value of <r (^ + 2ma)i + 2n<»9), where m and n are any integers. An example shemng how the function o- {z) may he expressed as a singly-infinite product We have <r («) =zn (1 - ^ ?-« I ea»«,+ai»«,"*'^»m«,+2«-,)« , the Biunmation being extended over all positive and n^ative integer and zero values uf m and n, exoept simultaneous zeros. This can be written in the form »»±i V 27»«i/ ^.Ai \ 2w«8/ ±00 • / » \ * X m~±l jissl \ 2w«i + 2w«)2/ n n (1+- ^ )ea»-i+2n4./»(2m«»+2iu.J«. «i»±l n=l \ 2m«, + 27ki)j/ Now ±00 =.±l\ 2WKtt,/ IT 2«, and n n fl-5 —; — )e2«i«i+an«i4"*"H2»i4#,+amJ _«\ -.?«?izf \ 2nu»J ^ 2mwi ^ sm^- = %• f -2««tg 1 1 *^ 1 = n ^<»i ^ ^—Ai 12««, (2m«,+am»,)"*"*(a«M.i+2ii^/ 2<0j 29ta>2 Similarly *« « / ^ \ ^* — +1 ^ n n ( 1+^5 I «a»t«i+2«Wi4 (2»n«i+2»M*a)* m»±i «=i \ 2mtt>i+2w«2/ . (2nci>a + z)ir ^ « ^ ^ sin^-— I ^— ^ f 2w»,8 +1 '^ \ sr n ^<^i ^ ^««±a8ti»«, (2iiu»,+2iM.^^'(2»iii»j+aiM^«; -1 8in?^ 1+-^ * ' 2a>j 2na>2 213] ELLIPTIC FUNCTIONS; GENERAL THEOREMS. 371 Therefore *• 1 2a», Sin 00 X n «=i 3 . 1 ^^ sin ^ ^ ^' ^ *j / -2»M,r Sin 2n<k)o7r Sin ~^-^- 2a»i (2mM,+2i»w9) 00 X n »=i e z£ 2a>, +j ^ sin 1 27K04S' sm '^ 6 ** r 2nMt' , . «* (2mM,+2n«08) (2mw,+2n«tft)' } 2a) 1 or ±m 1 «*JU ~ olu — ,_ L 5 z 2tt)i 2a>t g(2ii«,)« «=4i(2m«,+2»«,)« Sin*'^ ®1 Now write q^ e »! . Then .l„(^l^^.g!!fb±£); {,.,»-.-.g} {.. j!'-"' } . onfl»a7r Sin" — *- «1 { 2ltoaii r| m 1-25** 008 — hg^* Now if the imaginary part of tojta^ is positive, we have | ^| < 1 ; and thus the infinite product 1 - 2a*» cos — + ^* «=i (i-g^**)^ converges ahsolutely, since the series 00 n=l converges absolutely ; and hence we can separate off the exponential factors, and can write where (7 is a constant. l-2^cos-- + a*» «1 The quantity (7 can be very simply determined from the relation «r(«+2«i)=-c^''»(*+"»)«rW; for this gives or (7=-5>~ 2<k>, 872 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI. We have therefore finally an expression for o- (2) as a singly-infinite product, namely where q=€ •*! . 214. 2%6 integration of elliptic functions. The integral of any elliptic function can be found in terms of the functions f(«) and (r (z), by using the theorem given in §211, on the resolution of elliptic functions into a sum of (['-functiona In fact, in § 211 an expression has been found for the elliptic function f(z) ; the indefinite integral of this • • expression is c-^+ 2 c*. log o- (2: - aik) + 2 2 7 YTf <^*. f^*"*^'^^ - «*)> *=i *=i #=2 V*— 17' which is the required integral of f(z). Example, The expression for fp^ {t\ found by the theorem of § 211, is It follows that jii^(z)dz=ifp^(z)+ ^g^ + Constant. 216. Expression of an elliptic furvction whose zeros and poles are knoum. We have already seen (§ 205) that the number of irreducible zeros of an elliptic function is equal to the number of its irreducible poles ; and that (§ 208) the sum of the aflSxes of the zeros differs from the sum of the affixes of the poles only by a quantity of the form {^nitoi + 2rko,), where m and n are integers. By replacing the zeros and poles by others congruent to them, we can reduce this difference to zero. Suppose this done, so that for a given function f{z) the irreducible zeros are Oi, a,, ... On, and the irreducible poles are 61, 6,, ... h^ where Oi + Oj + . . . + On = 61 + 61 + . . . + 6n« If any of the zeros or poles is multiple, of order k say, it will of course be counted as if it were k distinct simple zeros or poles. Consider now the quantity ^ / X ^ a{z-a^)a{z'-a^) a{z-a^ ^ a{z — b^a{Z'-b^ a{z — bf^' 214,215] ELLIPTIC FUNCTIONS; GENERAL THEOREMS. 373 ^ We have Similarly J? (z + 2w^) = E (z). Thus E(z) is an elliptic function, with the same periods aa /{z); and therefore f{z)/E{z) admits these periods. But the function f(z)/E(z) clearly has no zeros or poles at the points djf eta, ... ttnj ^i> ••• Ofif and so has no zeros or poles at any point of the 2r-plane. Therefore, by Liouvilles theorem, f(z)/E{z) is a constant; and so finally ''' ^'^''a{z^b,)a{z-'b,) o-(ir-6n)' where c is some constant. An elliptic function is therefore determinate, save for a multiplicative constant, when the places of its irreducible zeros and poles are known. This is analogous to the factorisation of a rational function : if a rational function has zeros at points a^ a,, ... o^, and poles at points &|, b^y ... b^, it can be expressed in the form (g-at) (g-ag) ... (z-On) U^-b,)(z-b^,..(z-b^y where c is a constant. Example 1. Prove that By differentiating this formula, shew that and by further differentiation obtain the addition-theorem J>(^+y)=-f>W-j>(y)+;{ ^g:g'g} V Example 2. If 2 (aA-6A)=0, shew that S «^ («A- ft.) ... o- («A- 6a) ■■■ .r (ax - 6^^^ A=l (r(OA— a,)... 1 ...<r(ax-o») 24—3 374 TRANSCENDENTAL FUNCTIONS. [chap. XVL Miscellaneous Examples. 1. Shew that, ifp denote one of the functions an e, cnZf dn z, and if q and r denote the other two, it is always possible to choose constants a, 6, c, such that / jxiz = a log (bq +cr). 2. Shew that every elliptic function of order n can be expressed as the quotient of two expressions of the form where 6, Oj, a,* ••• ^n* ^^ constants. (Painlev^.) 3. Prove that ip(^-a)if>(^-6)-if>(a-6){if>(2-a) + fr>(^-6)-i?(a)-if>(6)} + j?(«)P(&). (Cambridge Mathematical Tripos, Part II, 1895.) 4. Shew that <r (x-k-y+z) a (x-y) <r (y - z) <r {z - a;) _1 1 i?(^) rw 1 j?(y) r(y) Obtain the addition-theorem for the function fp (z) from this result 5. Establish the identity 1 PW i»'(^x)...««-"(^i) ^^^" ^"^ 1 if>W jf>'W...««-*>(0 . where the product is taken for all int^er values of X and ft from to n, with the restriction 6. Prove that f(-?-a)-f(2-6)-f(a-6) + f(2a-26) <r {z^2a-\-b) a {z- 2b-{-a) <r{2b — 2a) <r(z-a) o-(z — b)' (Cambridge Mathematical Tripos, Part II, 1896.) 7. Shew that, if Zf^+Zi+Z2+z^=0y then {2f (z^)}'=3 {Sf (2a)} {2fp (2a)} + 2P' (2a), the summations being taken for X=0, 1, 2, 3. (Cambridge Mathematical Tripos, Part II, 1897.) MISC. EX8.] ELLIPTIC FUNCTIONS ; GENERAL THEOREMS. 375 8. Prove that f^^ (r(Z'^Zi)(r{z-¥22)<r(z+z^)fr(z+z^) is a doubly-periodic function of z, such that = ~2a-{i(«8+28- 2,-^4)} <r{i(«3+2i-«,-«4)}(r{i(«i+28-a3-«4)}. (Cambridge Mathematical Tripos, Part II, 1893.) 9. If /(z) be a doubly-periodic function of the third order, with poles at «=0i, z^Cj, -?=c„ and if ^ («) be a doubly-periodic function of the second order with the same periods and poles at 2= a, 2= /3, its value in the neighbourhood of 2= a being <(,(£)= A-+Xj(2-a)-|-X2(^-a)*-H..., prove that i X« {/" (a) -/" (/3)} - X {/' (a) +/' 03)} 2 * (c,) + {/(a) -/(/3)} {ZXk, + 2 <^ (c,) <^ (c,)} -0, (Cambridge Mathematical Tripos, Part II, 1894.) 10. If X(;s) be an elliptic function with two poles a^ a,, and if ;?i, ^, ... z^^ be 2n arbitrary arguments such that shew that the determinant whose nth row is 1, X(«<), X«(^<),...X-(«<), \{Zi\ \(zi)\{Zi\ X«(^)Xi(^,),...X-2(z<)Xi(z<), where \(^)=^X(«<), vanishes identically. (Cambridge Mathematical Tripos, Part II, 1893.) 11. Shew that, provided certain conditions of inequality are satisfied, \ f\ e "I =5— (cot— --hcot-^)-h — Sg^'^sm — (»iw+»y), where the summation applies to all positive integer values of m and n. (Cambridge Mathematical Tripos, Part II, 1895.) 12. Assuming the formula 'W=* ' • V "" 2i, ? — (TTj^i — > prove that on condition that (Cambridge Mathematical Tripos, Part II, 1896.) 376 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI. 13. Shew that 26 1 ^W-6(«-6) (Dolbnia.) INDEX OF TERMS EMPLOYED. ( The numbers refer to the pages, where the term occurs for the first time in the hook or is defined.) Absolute convergence, 12 ,, valne (modulus), 5 Affix, 6 Analytic function, 46 Argand diagram, 6 Associated Legendre functions, 281 Asymptotic expansion, 163 Automorphic functions, 389 Beraoullian numbers and polynomials^ 97 Bessel coefficients, 266 „ functions, 274, 294 Branch, branch-point, 66 Circle of convergence, 29 Coefficients, Bessel, 266 Complex numbers, 4 Conditions, Dirichlet's, 146 Congruent, 325 Contiguous, 260 Continuation, 57 Continuity, 41 Contour, 47 Convergence, 10 absolute, 12 circle of, 29 „ radius of, 29 semi-, 12 uniform, 73 Cosine series, 138 Definite integral, 42 Dependence, 40 Derivate, 51, 53 Determinants, infinite, 35 Diagram, Argand, 6 Dirichlet*8 conditions, 146 „ integrals, 191 Double-oircnit integrals, 258 Doubly-periodic, 322 ♦» ft I* }> i> »» ♦» »» Elliptic function, 322 Equation, associated Legendre, 231 Bessel, 269 hypergeometric, 242 Laplace's, 311 Legendre, 206 Essential singularity, 63 Eulerian integrals, 184, 189 Expansion, asymptotic, 163 Exponents of a singularity, 245 Fourier series, 127 Function, analytic, 45 associated Legendre, 231 automorphic, 339 Bessel, 274, 294 Gamma-, 174 elliptic, 322 hypergeometric, 242 identity of, 59 Legendre, 209, 221 many-valued, 66 n »♦ II 11 II II II II Gamma- function, 174 Genus, 339 Geometric series, 13 Hypergeometric series, 20, 240 „ function, 242 Identity of a function, 59 Infinite determinants, 35 products, 81 series, 10 II i» Infinity, point at, 64 Integrals, definite, 42 Dirichlet's, 191 double-oircnit, 258 Eulerian, 184, 189 Invariants, 326 II II II 378 Irreducible, 362 Kind of Legendre functions, 209, 221 „ Bessel „ 274, 296 Laplace's equation, 811 Legendre associated functions, 231 equation, 206 functions, 209, 221 polynomials, 204 Limit, 8 INDEX. »i i» i» Many-valued function, 66 Modulus, of complex quantity, 5 „ Jacobian elliptic functions, 346 Non-uniform convergence, 73 Numbers, BernouUian, 97 „ complex, 4 Order of Bessel coefficients, 267 „ functions, 274 elliptic functions, 368 Legendre functions, 209 „ polynomials, 204 pole, 63, 65 zero, 55, 64 >» a »» i» ti II Parallelogram, period-, 825 Part, principal, 63 Period, 322 Period-parallelogram, 325 Point, regular, 45 ,, representative, 6 „ singular, 45 Pole, 68, 66 Polynomials, BernouUian, 97 „ Legendre, 204 Power-series, 28 Principal part, 68 Process of continuation, 57 Products, infinite, 81 Quantity, complex, 4 Radius of convergence, 29 Regular, 45, 46 Residue, 83 Representative point, 6 Semi-convergence, 12 Series, Fourier, 127 geometric, 18 hypergeometric, 20, 240 infinite, 10 power-, 28 sine and cosine, 188 Simple pole, 63 Sine series, 138 Singly-periodic, 322 Singularity, 45 „ essential, 68 of hypergeometric equation, 245 II If II II II II Uniform convergence, 73 Uniformisation, 388 Value, absolute (modulus), 5 Zero, 65, 64 CAMBBmOE ; PRINTED BY J. & C. V, CLAY, AT THE UMIVBBSITT PRB88. fl t w '•■^1 IF